Critical behavior in neural networks is characterized by scale-free avalanche size distributions and can be explained by self-regulatory mechanisms. Theoretical and experimental evidence indicates that information storage capacity reaches its maximum in the critical regime. We study the effect of structural connectivity formed by Hebbian learning on the criticality of network dynamics. The network only endowed with Hebbian learning does not allow for simultaneous information storage and criticality. However, the critical regime can be stabilized by short-term synaptic dynamics in the form of synaptic depression and facilitation or, alternatively, by homeostatic adaptation of the synaptic weights. We show that a heterogeneous distribution of maximal synaptic strengths does not preclude criticality if the Hebbian learning is alternated with periods of critical dynamics recovery. We discuss the relevance of these findings for the flexibility of memory in aging and with respect to the recent theory of synaptic plasticity. Critical dynamics in neural networks is an experimentally and conceptually established phenomenon which has been shown to be important for information processing in the brain. Critical neural networks have optimal computational capabilities, information transmission and capacity  of the synapse connecting j to i. Action potentials are generated and delivered to all postsynaptic neurons when neuron i reaches the membrane potential threshold θ. After this depolarization, the potential is reset according toThe first term on the right hand side of Equation 1 represents an external excitatory input of size J = {Jij}. For a fully connected network with equal weights the activity forms a series of avalanches that are separated by longer periods of quiescence  of successive depolarizations. Because some of these activations may occur simultaneously, avalanches are also characterized by their duration (D), i.e., the time from the start of the avalanche to the firing time of the last neuron that was activated in this way.The activity dynamics in this model network depends on the connectivity and the weight matrix Jcr exists that leads to a scale-free avalanche size distribution P(L)  and the best-matching power law over the range 1 ≤ L ≤ N/2. Unless Δγ is not less than a specified threshold Δγmax, we keep recording Aava avalanches until the resulting size distribution has converged toward a power law and this sets the end of the critical regime. The resulting synaptic weight configuration {Jij} does then represent a neural network operating at the critical point. For small network sizes Δγ was shown to be as informative about criticality in the network as a Kolmogorov-Smirnov statistic with Monte-Carlo generated p-value  and Jij(t + 1) are respectively the old and the new synaptic strengths and εhebb ≪ 1 is a learning rate. Note that we do not apply Equation 7 synchronously for all the synapses but rather in a stochastic manner with update probability p = 1/N for each synapse . This is done until the synaptic weight configuration {Jij} allows for associative recall of the stored patterns as specified below. We will refer to the episode during which Equation 7 is applied as Hebbian learning.Here, A modified learning rule is implemented in the case of dynamical synapses, which is given byTij} but rather to the maximal efficacies Tij} tend to closely recover to their maximum values and these do not contain information about the stored patterns. If, however, the Wij}, the instantaneous efficacies {Tij} will be affected in favor of the memory configuration too because they recover during episodes of low network activity toward the Unlike before, Hebbian learning is not applied to the instantaneous values of synaptic efficacies {Clearly, we need a criterion that sets the end of the Hebbian learning episode. This criterion can only be based on the retrieval quality of the current network state, which is discussed in the following section.Jij(t)}, we construct perturbed versions κη = Qξη of the stored patterns. The operator Q selects an active and an inactive neuron at random and swaps their states, thereby keeping the total number of active neurons unchanged. Ideally, the network will be able to reconstruct the original ξη from the κη using the information that is implicitly stored in the connections. Practically, we can only require that the network produces a state that has less errors than κη, i.e., that is closer to the stored pattern than the perturbed version.In order to assess the retrieval (or memory) quality in the network with the configuration of synaptic strengths {η, the network activity will switch to a new configuration Sη given bySη as the retrieved pattern. To quantify the overlap between two binary patterns ξ and κ we use the correlational measuremξ and mκ denote the mean activities and σξ and σκ are the standard deviations, respectively. Perfect overlap is obtained for identical patterns where we have o = 1, while we obtain o = 0 for uncorrelated patterns. Thus, the observation thatSη, ξη)〉ξ compared to the perturbations, provides evidence that the weight configuration {Jij} indeed contains information about the stored pattern ξη. If the network realized, in contrast, an identical transformation it would not achieve an improvement of the overlap, but it could “remember” a pattern for a short time in a kind of short-term memory. Typically, we will not only consider a single random perturbation κη per pattern but many, so that Equation 11 becomesIn discussing these questions, we will use a simplified model which was chosen mainly in order to be able to relate to results in Levina et al. , 2009 asIn a spiking network also temporal averages need to be included in order to obtain a consistent measurement of the retrieval quality. According to the simplifying assumptions above, we will consider only small perturbations which consist in the case of a finite network in single bit swaps. This is done for two reasons. First, such perturbations are used in order to concentrate on the threshold-independent effects of the retrieval dynamics. A perturbed pattern cannot be corrected by the choice of a standard threshold value. Second, near the critical capacity, it is sufficient to study the ability of the network to correct a single error. This is due to the reduction of the size of the basins of attraction of the pattern-related attractor states. As soon as the attractor size has reached zero even an almost correct pattern will typically deteriorate with the dynamics (Equation 9). A persistence of a fixed point state beyond the capacity limit, but without a basin of attraction, is easily achieved, e.g., by avoiding any update of the neurons state, but is not interesting in the present context.Two comments concerning the threshold Θ in Equation 9 seem to be in order here. First, we choose this threshold such that the average overlap hebb. For the sake of reduced numerical complexity, we only consider one single perturbation for each stored pattern, i.e., we use Equation 11 instead of Equation 12. After Hebbian learning is over, the network is driven toward the critical regime again, employing either homeostatic regulation of synaptic weights or dynamical synapses, respectively. This alternation between the two episodes may be interpreted as an optimization scheme and the delicate question is if convergence toward a state is obtained in the long run, in which the network is critical and retains an associative memory of the stored patterns at the same time. In what follows, we will refer to these states as converged states.In the previous sections we have outlined how the synaptic weights evolve during the critical and Hebbian learning episodes, respectively. The critical episode ends as soon as the sampled avalanche size distribution is close to a power law (see section 2.3), whereas the following phase of Hebbian learning is stopped as soon as the network exhibits good retrieval quality of the stored patterns, when perturbations of the latter are presented. We measure the retrieval quality after each step of Hebbian learning and stop if the improvement in average overlap is at least Δconv and take the average over np (np ≫ 1) perturbations per pattern. Note that in the case of dynamical synapses, we assess the retrieval quality of Jij = uTmaxij in the Hebbian learning phase, whereas we take Jij = uTij to check for convergence.Whether the network is in a converged state is always checked after the critical episode is finished and before the next round of Hebbian learning is started. At this point we already know that the system is operating at the critical point but we still need to make sure, that the critical episode has not erased memory of the stored patterns. We therefore rigorously test the retrieval quality of the network using Equation 12 with a minimum required improvement of ΔA sketch of the optimization strategy is shown in Figure N = 300 neurons. Other parameters are summarized in Table In this study we always simulate networks of A0 avalanches without taking them into account in the avalanche size distribution P(L). This is done to ensure that the size distribution is not affected by transient dynamics. The sampling of the distribution is an important contribution to the total simulation time and was the main limitation of the size of the network. Because in larger networks also larger avalanches need to be considered, the sampling time for given Δγmax increases faster than quadratic which was the main reason for our choice of the network size. Smaller networks, however, are less suitable to store small activity patterns, see section 3.2.At the beginning of each critical phase we sample M stored in the network, where each trial uses a different set of M patterns. Unless otherwise stated, data points are averages over 10 trials for each M and error bars indicate one standard deviation from the mean. Instead of the number M of stored patterns in the network, we will typically use the load parameter, defined as α : = M/N.Apart from that, we consider several trials for each number of patterns M random binary patterns, calculate the matrix {Wij} according to Equation 6 and set the network connectivity to Jij = Wij. The memory quality is then assessed by calculating the average overlap Sη and the original patterns ξη (see section 2.5 for details). From here on, we always take the average 〈.〉κ over np = 1000 randomly generated perturbations κη of each of the M patterns ξη. The average overlap is close to 1 up to load parameters α ≈ 0.07  from the best-fit power law. Although there is no mechanism in these networks to bring about criticality in a self-organized way, we always choose the normalization C in Equation 6 such that 〈Wij〉 corresponds to the critical value of the model with fixed synaptic weights [see section 2.1 and Eurich et al. −1. Due to Hebbian learning however, structure in the maximal resources will develop in the course of the simulations.We now turn to the second synaptic regulatory mechanism that brings about criticality in our networks (see section 2.3). All the analysis in this part is done along the lines of the previous section, so the only essential difference here is that we substitute homeostatic learning as the mechanism that drives the network into the critical regime by dynamical synapses. At the beginning of the simulations, maximal synaptic resources Also the model networks considered here evolve toward converged states that are critical and an associative memory at the same time. While the total number of Hebbian learning steps needed to converge Figure  is compao〉ξ at α ≈ 0.13  memory quality in the converged state, we presented perturbed versions of the initially stored patterns to the networks which resulted in retrieved patterns. The latter almost never deviate from the original patterns more than the perturbed states themselves. More importantly, more than 90% of the patterns are reconstructed with on average less than one digit deviation from the original patterns. We may therefore conclude that the memory quality of the critical networks is very good. However, a pure memory network which has couplings equal to {We have considered two mechanisms that can regulate a neural network toward criticality . Homeostatic learning regulates synaptic weights until the branching ratio approaches the critical value. Dynamical synapses on the other hand represent a biologically more justified regulatory mechanism, where the critical branching ratio is reached through the interplay of synaptic depression and recovery. In the latter model, we found that agreement between criticality and memory can only be achieved if the maximal synaptic weights are structured by Hebbian learning. Homogeneous maximal weights in contrast lead to memory loss during critical episodes because synaptic resources may recover to their maximal values which carry no information of the stored patterns anymore.The apparent contradiction between the classical concept of memory storage by fixed synaptic efficacies which are modifiable only by persistent high-rate stimulation on the one hand, and the realization of adaptive filters based on the short-term dynamics of synaptic resources has been studied already in Tsodyks et al. , BibitchOne of the main points here is that the critical state serves as a ground state of the system which is assumed in the absence of specific external input. But it is, since the completely inactive state is absorbing in our model, constantly fed by spatially and temporally homogeneously distributed external noise. A specific external input has a large overlap with one of the patterns and a small overlap with all the other stored patterns. This is a necessary condition of the model, which in order to be relaxed requires a specific modification. We have dealt with such problems in our previous papers, but assume here uncorrelated patterns. In this way the overlap between any two patterns is negligible in theory. In a finite network this is not necessarily the case, but for a limited number of patterns the overlaps are smaller than the threshold for the spill-over into any of the other patterns. In a memory network below the capacity limit, the activity will therefore be confined to the pattern which is indicated by the input. The dynamics will thus not be critical. In a critical network on the other hand, we conjecture that on short time scales avalanches will be correlated to the existing patterns. Since, however, all neurons and thus all patterns receive constantly external input now, the avalanches are not confined to a pattern but will jump into other patterns on medium time scales.Although we have not made this explicit here, the memory test can be formulated as an informational criterion (Rieke et al., The network considered here can be characterized by the interplay between the attractor dynamics in memory retrieval and critical dynamics that provides optimal exploration of the state space. In a system like the human brain, where the number of memories increases for a large part of the personal history, it seems that there must be eventual a consequence for the flexibility and the ability to explore new patterns (Schrobsdorff et al., Although we could show here that memory storage and criticality are not irreconcilable, our results support a view that has been adopted by an increasing number of researchers in the last decade, namely that memory traces are not necessarily point attractors but more general dynamics objects (Herrmann et al., We have demonstrated that criticality can be preserved in an attractor network if both a memory consolidation process and a mechanism for regulation toward criticality are present. Among the different mechanisms for the maintenance of a critical regime, we found that the dynamics of synaptic resources is both biologically realistic and effective for criticalization, while their effect on memory capacity is moderate. Other mechanisms are possible, but are less easily biologically justifiable and have eventually a disastrous effect on the memory content unless actively counteracted. It is necessary to consider the present results in more realistic network models and under more general learning paradigms in order to understand better their significance for the natural and pathological development of biological neural systems.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


We present a new mechanism for detecting shared bottlenecks between end-to-end paths in a network. Our mechanism, which only needs one-way delays from endpoints as an input, is based on the well-known linear algebraic approach: singular value decomposition (SVD). Clusters of flows which share a bottleneck are extracted from SVD results by applying an outlier detection method. Simulations with varying topologies and different network conditions show the high accuracy of our technique. End-to-end flows in packet based networks adversely influence each other when they traverse a network link which is a single point of congestion. We call such a link a “shared bottleneck”. Given the large number of distributed applications that are now used on the Internet, where coordination between such end-to-end flows is quite feasible, we believe that the detection of shared bottlenecks in the network is one of the key measurement efforts that can be undertaken, as there are numerous possibilities for using such knowledge.Consider, for example, a transfer delay prediction tool  which tells two senders A and B that the time to transfer files to receivers C and D will be four and five minutes, respectively, but fails to inform them that the A-C and B-D path share a bottleneck. Then, both systems would perhaps start to send the files at the same time, and the predicted transfer delay would clearly be wrong because of adverse interactions between the flows at the shared bottleneck. In a Grid, it is not uncommon for a single entity (or person) to be in control of the two senders A and B—thus, with knowledge about shared bottlenecks, this problem might have been prevented by accordingly scheduling the time of these transfers (this is what schedulers in Grid workflow systems do).As an another example, one could imagine an overlay network where a file is sent from a single source to a single destination via multiple paths across the overlay in order to speed up the transfer time. Such a scheme would probably only yield a noticeable benefit if the overlay paths that are used do not include a shared bottleneck in the underlay. Similarly, in peer-to-peer systems, knowledge about shared bottlenecks could theoretically be used for appropriately rearranging the way documents are placed in order to exploit multiple paths or avoid adverse interactions among simultaneously active downloaders.To the best of our knowledge, despite the existence of some measurement studies on shared bottleneck detection cf. –3), such, such3])In this paper, we address this issue with a method that only operates on end-to-end measurements (forward delays), is easy to calculate, and provides extraordinarily accurate results. Our scheme is based on a combination of the well-known technique of singular value decomposition (SVD) of principal component analysis (PCA) and outlier detection.After a brief survey of related work in the next section, we will provide the mathematical foundations of our mechanism in The awareness of the complete network topology and particularly the knowledge of congested network links which are shared by multiple flows can be of great use to many applications. A great deal of work has already been done in this area which directly or indirectly relates to our study. In what follows, we briefly discuss some of the related efforts.Two techniques have been proposed in  which arIn the delay based technique, the cross measure is computed by merging the delay sequences of the two flows which are to be tested. From this merged sequence, only those pairs of samples are picked in which the first sample of a pair is from one sequence and the second sample of pair is from another sequence. Then, from all such samples, the cross measure is calculated between the samples of the first and the second sequence. The automeasure is calculated as the correlation coefficient between the delays of adjacent packets of a single flow. After the computation of the cross measure and the automeasure, the comparison test declares that the two paths share a bottleneck if the automeasure is less than the cross measure.In the loss based technique, the cross measure and the automeasure are actually the conditional probabilities that a packet is lost given that the following packet is also lost. Both techniques work in the topologies where the senders or receivers are colocated. It is also shown that the delay based technique attains accuracy quite quickly as compared to the loss based technique.The entropy based technique presented in  works inWe consider FlowMate  as one oOne of the latest works in this regard is by Kim et al. , where aOne thing which is common in all the above-mentioned approaches is that all of them operate in a pairwise fashion. All techniques consider two paths to detect if flows share a bottleneck whereas our mechanism works for any number of flows at the same time, resulting in clusters of paths which share a bottleneck.A loss based technique which expects a common sender to send packet pairs has also been proposed . If the Multivariate analysis has most of its applications in image processing and pattern recognition, but the nature of our problem makes multivariate analysis an ideal tool in solving it. We decided to apply multivariate analysis because we believe that there are many factors which are beyond the scope of a simple correlation method. Multivariate analysis is highly suitable in such scenarios because we want to check the principal factors which mean the delays caused by the congested link.We will see that SVD provides us with highly accurate results in terms of identifying the flows which share a bottleneck. SVD is one out of many multivariate analysis techniques in the literature, and others should eventually also be considered in terms of efficiency and reliability.A in which the rows are the flows and the columns are time slots. An entry in A is the amount of (average) delay measured during experiments. We will analyze this data matrix A using SVD.We construct a matrix We briefly review the singular value decomposition of matrices; we will use some of its properties in our discussion in m × n matrix A can be expressed asr is the rank of A, σ1(A) ≥ σ2(A) ≥ ⋯σr(A) > 0 are its singular values, and ut)('s and vt)T(uj) matrices of dimensions m × r and n × r, respectively, containing the left and right singular vectors of A. Σ = diag(σ1(A) ≥ σ2(A) ≥ ⋯σr(A)) is an r × r diagonal matrix containing the singular values of A.Any A = ∑t=1rσt(A)ut)(vt)rut)(vt)r for various values of k. Essentially, discarding the “smallest” principal components  results in a small loss in accuracy, and we might justifiably consider those components as “noise”. This procedure is commonly referred to as “principal component analysis,” and the following theorem  quantifAk = ∑k=1rσt(A)ut)(vt)r( (for any 1 ≤ k ≤ r). Ak is the “best” rank k approximation to A with respect to the 2-norm and the Frobenius norm; namely, for any matrix D of rank at most k,A has a “good” rank  k approximation if the 2-norm and the Frobenius norm of A − AK are small; for a detailed treatment of singular value decomposition, see [Let ion, see .The main intuition behind using SVD based approach was obtained from the work of Camtepe et al.  where thmeasurement of one-way delays along all the paths and obtaining a covariance matrix,applying SVD on the covariance matrix and getting a projection,grouping paths that share a bottleneck.A bottleneck is a network link/router where the packets of a flow experience excessive delays or they are even lost. Most of the techniques proposed till now to detect shared bottlenecks are based on packet delays or packet loss. It has been shown that delay based techniques quickly converge towards accurate results ; hence, First of all we measure the one-way delays for all the paths by initiating a flow on every path. We group samples of delay measurements on the basis of time windows and pick the average of a time window as a representative delay measurement for that particular time window. We have worked with different time windows and observed no significant change in our results; this is explained in D of forward delays of all flows as shown in  and overhead . We can also reduce or expand the time window size depending on the delay samples available; for instance, in case of short lived flows, we can reduce it, and, in case of long lasting flows, we can expand it, for example, if we need to work in a fully passive environment. At the end we form a matrix shown in  where eafined in :(6)aij=Here, it is important to note that, if clocks are not synchronized between sender and receiver, then all the delay measurements will be shifted by a certain amount. The shifting will not affect the result of SVD based approach because the important factor for the result is the delay pattern and not the magnitude of actual delays, so the lack of time synchronization will only cause a shift in actual delay values, but the delay pattern will still be preserved. Am×m of delay measurements, we apply SVD on Am×m to get singular values of our delay measurements for all the paths. As a result we get the left singular matrix Um×r, the diagonal matrix Σr×r, and the right singular matrix Vn×r. Here, the matrix Um×r represents the association among the paths on the basis of similar delays caused by common bottlenecks, and the matrix Σr×r represents the degree of this association. Then, on the basis of rank reduction of the diagonal matrix Σr×r, we project the matrix Um×r on the c column(s) (for  any  1 ≤ c ≤ r) of the diagonal matrix to look for c dimensions. The value of c can be decided by looking at the explained variance of the elements of the diagonal matrix. For instance, if the first element of the diagonal matrix is explaining the data 60%, then we may need to take the second or third dimensions. If the first one explains 90% of the data, we say there is rank reduction and project on the first column of the diagonal matrix to take only the first dimension.Having the covariance matrix Um×r on the cth column of the diagonal matrix, we multiply each row of Um×r with only the cth column of the diagonal matrix. This results in a matrix Xm×1, as shown in  between two successive values of the sorted list by usingτ defined ass is its standard deviation. The coefficient value of 0.2 was selected after a detailed study of data obtained in simulations. As we will see in In order to detect outliers, we used the threshold It is important to have a computable time complexity for situations where a mechanism needs to be applied in real time. In this section, we discuss the time complexity of our algorithm and we show that it is feasible to use online. It has three major parts. During the first part, we extract singular values from delay measurements, then in the second phase we find the projection of left singular vectors over the first singular value, and finally in the third and last phase we identify clusters of paths that share a bottleneck.O(mn2) for a matrix of order m × n [n is the number of delay measurements used to detect shared bottlenecks. In O(mn2) is not expensive at all when we have an upper bound on n, so if we consider n as a constant, then the complexity becomes O(m), which means that it depends only on the number of paths under investigation. It is also true if covariance matrix is used instead of simple delay measurement matrix.First of all let us discuss the time complexity of SVD which is er m × n . Here, tO(mk) [k is the size of the data set.In an online version, we may need to update our results in a continuous fashion because the bottleneck link may vary depending on the load of network traffic. Solving SVD for one matrix of delay measurements allows for an incremental, efficient solution for letting the matrix slide over in time . In such situations, the time complexity of SVD is further reduced to O(mk)  where k U with the first element of the first column of matrix Σ. This procedure takes m multiplications which means a time complexity of O(m).The second phase involves the projection which mainly multiplies the first element of each row of matrix m values corresponding to each path as a result of second stage. In the final part of our procedure, we first sort these values, which may take O(mlogm) steps . At the end, for grouping paths which share a common bottleneck, we need to compare the sorted list with a threshold. This can be done in O(m) steps because it just involves m comparisons. We perform all above steps in serial so the most dominant part defines the time complexity of our whole algorithm. The total time complexity is, therefore, O(mlogm) where m is the number of paths to be inspected for common bottlenecks. This shows that it is practical to use this approach in real time.We get In this section, we present the results of using our mechanism in simulations. We used the ns-2  network The symmetric topology is shown in The asymmetric topology is shown in Three different performance metrics which can be considered for measuring the accuracy have been used in  as folloFalse Positive: a flow does not share a bottleneck, but it is considered as being part of a group that shares a bottleneck. (i) False Negative: a flow shares a bottleneck with a group but is not identified as part of the group. (ii) Correct Clustering: a flow is in the same group as flows with which it shares a bottleneck.(iii) N paths, Pc is the set of correct clusters, Po is the set of clusters as a result, nfs is the number of flows erroneously included in a resulting cluster, and sj is the number of subclusters of a correct cluster that was split into sj subclusters in Po, then AI is calculated asThe accuracy index (AI) presented in  takes alIn this section, we discuss the results of our experiments in different ways to highlight the accuracy of our SVD-based method for detecting shared bottlenecks.1) Symmetric Topology. To illustrate our whole mechanism, we first used the symmetric topology shown in V are shown; these are the results after projection of the left singular matrix U on the first component of the diagonal matrix Σ  Asymmetric Topology. We performed experiments with 12, 24, and 36 flows with random start time within the first second of the simulation using the asymmetric topology and found AI to be always 1. The results are shown in (3) Window Size. We used the asymmetric topology with 12 foreground TCP flows (representing 12 paths). In this experiment, we wanted to check the impact of the window size on the accuracy of our method. By window size we mean the time interval for which we combine all delay measurements and choose one representative which is the average of that interval. A window size of 0.3 seconds is working fine, in what follows; we justify the use of this value. We ran the simulation for 60 seconds and obtained results after setting different window sizes. It is possible that, if we keep on reducing the size of the window, we will reach a point when some windows may not have even a single measurement due to congestion, meaning we will not have even a single delay measurement for that particular window. This must be avoided because the SVD may result in inaccurate results if we use some blanks or imaginary values in the matrix of delay measurements. At the same time, we should not make the window size too large because this could blur the information about the correct bottleneck, and the time series of delay measurements of different flows may not show any association. against the window size in On the basis of the above observations we propose a general principle to choose a window size for any environment such that it may not yield a percentage of empty cells of more than 2%. In the rest of our experiments, we used a window size of 0.3 which worked fine with all the dynamic network conditions and environments that we simulated. Another strength of our approach revealed in the above experiments was the amount of data required to converge to accurate results. By increasing the window size to 30 seconds we had only 2 average measurements per flow. In such situation, we had a delay matrix of 12 × 2 and it resulted in only 2 values in the diagonal matrix Σ, but even with such a small number of values in the diagonal matrix we had perfect results after the projection step.4) Drop Policy. We observed some interesting results in simulations using the asymmetric topology with different drop policies like RED and DropTail. DropTail performed well and showed a very promising behavior as shown in  Buffer Size. Our additional tests included scenarios with different buffer sizes from 50 to 500 packets. The results showed no significant difference. With DropTail, the AI was always 1. We also computed the AI after every 5 seconds for all above scenarios, starting from second 1 of the simulation till its end. We found that the AI value was 1 throughout the simulations.(6) Without Common Endpoints. In our experiments so far, we have used topologies with a common source. It has also been the major assumption in related work to have a single common source or destination. Here, we evaluate our technique for scenarios with multiple sources and destinations. The topology used for these experiments is shown in  to always be 1. This is also shown in As a next step, we again generated 12 TCP flows but from random sources to random destinations; again, our SVD based mechanism resulted in a very nice rank reduction and then clearly showed the outliers to form the group of flows which share a bottleneck. The results are shown in As a final step, we generated three TCP flows from every sender to a randomly selected destination in each branch of the bottleneck. In this way, we generated 36 flows in total, generating foreground traffic from each sender to every branch of the bottleneck. Even with such a heavy foreground and background traffic our mechanism was stable and reliable enough to produce clusters of flows having common bottlenecks with an AI of 1, which is shown in (7) Fluctuating Bottlenecks. It is quite possible in real networks that heavy cross traffic may cause a link to behave as a bottleneck for an end-to-end path for a while, and then a routing change or some sudden bursts may cause another link to become the bottleneck. Therefore, in our next experiments, we evaluated the adaptability of our mechanism in case of fluctuating bottlenecks.The topology used for these experiments is depicted in In our simulation of 30 seconds we congested the first link for the first 12 seconds of the simulation and then shifted the heavy cross traffic to the second link. As a result, for the first 12 seconds, paths p1 and p2 shared a bottleneck, and, for the rest of the simulation, paths p1 and p3 shared a bottleneck. Initially we calculated the AI for every 3 seconds and found that it accurately detected the bottleneck shift around the 12th second of the simulation. Then, in order to check the adaptability of our mechanism at a fine grained level, we also calculated the AI for every 1.5 seconds from the 9th second of the simulation till the 15th second of the simulation because this was the time when the bottleneck shifted. This time we noticed a drop in the AI (suggesting an inaccurate clustering) around the 12th second, as shown in From this experiment we came to know that our mechanism was efficient enough to adopt the shift in the bottleneck within 3 seconds, which is quite acceptable because, if the bottleneck fluctuates every time within less than 3 seconds, then this information will probably not be very useful.(8) Topology with Longer Delays after the Bottleneck. This section highlights a problem that occurs if the delay matrix is used as an input for the shared bottleneck detection algorithm. This problem is not very common but may occur in some special scenarios—this is evident from all the experiments presented so far, where the delay matrix was used as an input for the shared bottleneck detection algorithm and no problem was visible in the results. A scenario where such problem might occur is depicted in Due to this setup, when forward delays are measured from the source to all destinations, then delays from the source to D2 and D4 become similar in terms of their magnitude. These magnitudes overlap so closely that this overlapping outweighs their delay patterns. This can be observed from the projection results presented in We compared our mechanism with FlowMate which, to the best of our knowledge, is the best of the mechanisms in terms of accuracy and computational complexity.1) Asymmetric Topology. We used the asymmetric topology illustrated in (2) FlowMate with Cumulative Data. We used the setup from Mx, and for the automeasure Ma it needs packets separated by a certain value t. So the main reason behind this experiment was to observe and compare the performance of FlowMate when it has more than the sufficient amount of data. Our SVD based mechanism was already proven to be good enough to provide accurate results with a small amount of data, and this experiment also showed that our SVD based mechanism can perform well even with a smaller amount of data. We calculated AI after every 10 seconds of the simulation time after applying both mechanisms on accumulated data. FlowMate performed better than before and resulted in improved results as shown in  and the simulation of varying bottlenecks, where we deliberately tried to find the limit of our mechanism's ability to adapt to dynamic network conditions. In the RED simulations, we were able to enhance the precision by projecting the diagonal matrix over second column of the left singular matrix.Clearly, this high reliability is the main strength of our mechanism. We believe that, combined with its operational simplicity, we may have reached a turning point with our scheme where shared bottleneck detection may no longer be a mere measurement exercise but can in fact become a key practical element of future network or transport layer mechanisms. In addition to extending the performance evaluation of our mechanism with real-life measurements, we intend to contribute to this change in several ways in our future work, where we will embed our shared bottleneck detection method in the transfer delay prediction system that was briefly mentioned in the introduction as an immediate next step.


One of the archetypal task manipulations known to depend on frontal-lobe function is reversal learning, where a dominant response must be overridden due to changes in the contingencies relating stimuli, responses, and environmental feedback. Previous studies have indicated that the lateral prefrontal cortex (LPFC), the lateral orbitofrontal cortex (LOFC), the anterior cingulate cortex (ACC), and the caudate nucleus (CN) all contribute to reversal learning. However, the exact contributions that they make during this cognitively complex task remain poorly defined. Here, using functional magnetic resonance imaging, we examine which of the cognitive processes that contribute to the performance of a reversal best predicts the pattern of activation within distinct sub-regions of the frontal lobes. We demonstrate that during reversal learning the LOFC is particularly sensitive to the implementation of the reversal, whereas the LPFC is recruited more generally during attentional control. By contrast, the ACC and CN respond when new searches are initiated regardless of whether the previous response is available, whilst medial orbitofrontal cortex (MOFC) activity is correlated with the positive affect of feedback. These results accord well with the hypothesis that distinct components of adaptable behaviour are supported by anatomically distinct components of the executive system. ► We model fMRI data at distinct stages of a reward driven reversal learning task. ► Lateral orbitofrontal cortex responds particularly strongly at the point of reversal. ► Lateral prefrontal cortex shows a similar response during other switches. ► Medial orbitofrontal cortex activity correlates with the rewarding value of feedback. ► Anterior cingulate cortex and caudate respond whenever new searches are initiated. The ability to alter behaviour according to changes in the environment is important for the survival of any organism. One of the classic measures of behavioural flexibility is reversal learning, where a dominant response is overridden in favour of a weaker competing alternative due to a change in feedback contingencies. Cortically, it is now well established that damage to the ventral prefrontal cortex leads to impairments in visual discrimination reversal learning in humans , monkeysThe reversal learning process, whilst providing a useful gauge of behavioural flexibility, is cognitively heterogeneous, being composed of multiple sub-processes. For example, successful reversal performance requires a participant to process task feedback and to instigate searching for an optimal behaviour by overriding a dominant response and reorienting the focus of attention. Thus, in order to better specify the neural basis of reversal learning, the present study sought to determine how anatomically distinct components of the executive system responded during the cognitively separable stages of the reversal learning process. In particular, we wanted to determine whether activations in both the LPFC and LOFC were specifically related to feedback driven learning in the presence of contingency change or not. Accordingly, participants undertook a series of visual discriminations and reversals in which they had to select one of a set of abstract patterns, subsequent to which they received feedback in the form of either an attractive or an unattractive female face. The reward contingencies were probabilistic and reversed unpredictably after a variable number of consecutive correct responses. This design enabled us to examine whether the BOLD responses were related to i) the processing of negative feedback in general, ii) the initiation of a new search iii) switching response from one object to another in the presence of a prepotent response  or iv) switching response from one object to another, in the absence of a prepotent response (acquisition). Such a comparison has not been possible in the majority of previous imaging studies of reversal learning, because these studies investigated either the serial reversal learning of a single discrimination  — thus lNineteen healthy right handed, male volunteers with no history of psychiatric intervention or neurological illness participated in this study . All participants gave written consent prior to taking part and the study was approved by the Hertfordshire Local Research Ethics Committee.On an individual trial, a stimulus set containing three abstract patterns was displayed  and the Each block, therefore, consisted of five distinct phases of behaviour. In the first phase ‘acquisition 1’ the participant explored the outcomes relating to selection of the different patterns. In the second phase ‘criterion 1’, the participant settled on selecting repeatedly the same pattern that they deemed most likely to lead to positive feedback. In the third phase ‘contingency change’ the rule defining which pattern was most likely to lead to positive feedback changed, and the participant determined, on the basis of the negative feedback, that a change in behaviour would lead to a more rewarding outcome. This third phase culminated in a reversal event in which the participant inhibited the dominant routine response to the previously rewarded pattern and started selecting one of the other patterns. The participant then entered the fourth phase ‘acquisition 2’ in which they explored the new outcomes when selecting the patterns, subsequent to which they entered the fifth and final ‘criterion 2’ phase of consistently responding to the pattern that led to the most rewarding outcome. Only those blocks in which all five phases were successfully completed were included in the event related fMRI analysis.In half the blocks, although three patterns were presented on the screen, only two were available for selection since one of them was considered inactive, denoted by being covered in small crosses. Although any responses to the inactive pattern were recorded they did not lead to any change of the display. These blocks alternated such that after participants had completed a block of trials using 3 available patterns, they would begin a block with only 2 patterns available. This manipulation was intended to examine differences in neural response associated with the degree of choice available to the participant at the point of reversal. However, as no significant results were associated with this manipulation it will not be discussed further here.The attractive and unattractive face stimuli were presented as colour images with direct eye gazes. To ensure similarity in emotional facial expression only faces with neutral to mild smiles were included in the stimulus set. Faces were scanned from print media, obtained from other researchers  or selec‘You will be presented with three patterns on the screen. Selecting one of the patterns will result in a picture being displayed on the screen and the type of picture will depend on the pattern you have chosen. There are no right or wrong answers in this task. Whenever a new set of patterns appears, you are asked to try them all and after that, you are free to choose which type of picture you wish to view. On some trials, you may notice that one of the patterns is covered in crosses. This means that the particular pattern is not active and trying to select it will not result in viewing a picture. In this case, please make your choices from the remaining two patterns.’Participants were presented with instructions, both verbally and in written form, immediately before commencement of the training session. Care was taken to ensure that participants were not explicitly made aware of the attractiveness of the faces they viewed.In addition, participants were verbally reminded: ‘When you see a new set of patterns, please select each available pattern at least once. This is to make sure you have sampled all of the available patterns. After this you may continue selecting patterns according to your own preferences. Please pay attention to the images in case things change as you go along’.Prior to entering the scanner, all participants underwent a short training session (~ 5–10 min) to ensure that they understood and were capable of performing the task. This training session involved performance of at least two complete reversals on a laptop computer outside the scanner. All participants underwent the same training task and the patterns and faces seen during this session were not used in the scanning tasks. Once the participants had completed the scanning task, they were asked to rate all the faces on a computerised VAS (as described above).19 participants completed the task, each of whom undertook two 18 minute sessions of scanning acquisition in a 3T Bruker Medspec s300 scanner at the Wolfson Brain Imaging Centre, Cambridge, UK. Two data sets were lost due to technical problems, leaving 17 full data sets in the fMRI analysis. A total of 1005 gradient echo T2-weighted EPI images depicting blood-oxygenation level dependent (BOLD) contrast were collected for each of two sessions and the first 18 volumes from each session were discarded to avoid T1-equilibriation effects. We collected 21 slices per volume with a slice thickness of 4 mm, inter-slice gap of 1 mm and a TR of 1.1 s. Resolution within each slice was 3.125 mm × 3.125 mm. Slices were angled away from the orbits to avoid signal dropout in the OFC due to magnetic susceptibility inhomogeneity. A T1-weighted structural MRI image was also collected for anatomical data.Data were processed using Statistical Parametric Mapping 5 . Preprocessing consisted of slice time correction, reorientation, correcting for participant motion, geometric un-distortion using phase maps , spatial11 regressors were entered into the individual participant's fixed effects analyses in SPM 5. The regressors were formed by convolving onsets and durations of the events of interest with the canonical haemodynamic response function. Two of the regressors were defined during the acquisition phases when the participants were exploring the reward contingencies associated with responding to the different patterns. These regressors were defined according to whether the participant selected the same pattern as on the previous response (acquisition stay) or switched to selecting a different pattern after receiving negative feedback (acquisition switch). A further three regressors were defined during the criterion phases and the contingency change phase, where the participants were repeatedly responding to the pattern that they had identified as being the most likely to lead to positive feedback. These were defined as repetitions during criterion on the basis of positive feedback (an attractive face — criterion positive), repetitions during criterion after the reception of probabilistic negative feedback (criterion negative), and repetitions in the contingency change phase after receiving negative feedback (contingency negative). Two regressors were included representing the first response of a new search after criterion. The first of these was the reversal, representing the stage in time at which the participant first selected a different pattern to the one consistently chosen during the criterion 1. The second was the first time that the new stimulus set was presented after criterion 2 and the participant selected a new stimulus from that set for the first time (stimulus-set change). These regressors were defined from the visual onset of the patterns until the time at which the participant responded. In order to identify those brain regions in which the BOLD response correlated with the perceived attractiveness of the faces, another four regressors were formed from the onsets and durations of the feedback events. The first of these was modulated according to the VAS ratings for each face in a linear manner, whilst the other three were modulated by the second, third and fourth order non-linear polynomials of the VAS scores to ensure that any non-linear correlations between reported score and BOLD signal change were accounted for.Data for these events were extracted from anatomically defined regions of interest (ROIs) using the MarSBaR ROI toolbox , which tAfter the neuroimaging procedure, participants were asked to rate all the faces contained on the database using a computer based visual analogue scale (VAS). Subjective ratings of attractiveness showed a significant difference between attractive and unattractive faces  indicating that assignment of these categories had been successful.Behavioural data were collected from those blocks in which the participant successfully completed all five phases of the block including the reversal. On average, participants made 4.8 ± 1.9 (mean & standard deviation) responses before settling on a routine response and entering the first criterion phase. In the reversal phase, participants received an average of 2.79 ± 1.2 negative feedback events before performing a reversal and entering the second acquisition phase. The average number of reversals included in the analysis per participant was 18.05 ± 2.19. Examination of trials to criterion using a 2 ∗ 2 repeated measures analysis of variance (ANOVA), in which the conditions were acquisition session (1 or 2) and number of patterns available (2 or 3), revealed that there were no significant main effects of session  = 1.98, p > 0.05) or available patterns  = 1.15, p > 0.05), nor any interaction between these two factors (F < 1).Mean response time (RT) data were calculated for the 7 types of response included in the first level models . A repeaWe predicted that areas involved in reversal implementation in particular would be more activated at the point of reversal when compared with the non-switch events that followed negative feedback and the other switch events that did not involve overriding a routine response. Consequently, group level analyses focused on activations associated with the non-switch events that preceded a reversal (contingency negative), the reversal, responses at stimulus-set change, and acquisition switches . First, Brain regions that were particularly active during the performance of a reversal compared with other switch events were identified using two further contrasts. ROI data were first contrasted for the reversal minus the stimulus-set change events. In both cases, a routine response had first been developed, criterion had been reached, and a new search subsequently initiated. However, at stimulus-set change the routine response did not have to be overridden, as the previously selected pattern was no longer available. The inferior orbitalis and the mid orbitalis, i.e. LOFC, in the right hemisphere were significantly activated for this contrast b. The coROI data were then contrasted for the reversal minus the acquisition switch events. In both cases, the participant switched away from a previously selected pattern that was still available after receiving negative feedback, however, in the reversal, the pattern that was switched away from was more habitual, having been repeatedly selected throughout the criterion phase. Significant activation was evident selectively within the inferior orbitalis ROIs. Notably, whilst the mid orbitalis ROIs followed the same trend this effect did not stand up to correction for multiple comparisons c. By conIntriguingly, these results suggest that the ACC and CN may be more active at both stimulus-set change and reversal when compared with acquisition switches  — an une5,80 = 26.2 p < 0.001), a sub-threshold main effect of hemisphere , and a significant main effect of condition . Notably, there was a large significant interaction between condition and ROI , supporting the view that different ROIs were sensitive to different task demands. There was also a significant ROI * hemisphere interaction  highlighting the right lateralisation observed in the whole brain analyses. There were no other significant interactions. To examine the question of whether LPFC and LOFC ROIs in particular were differentially sensitive to condition, data were averaged across ROI and hemisphere for the LPFC and the LOFC and then compared using a repeated measures ANOVA with factors of ROI (LPFC vs. LOFC) and switch condition. There was a significant main effect of condition  = 8.5 p < 0.001) and importantly, a significant interaction between ROI and condition  = 11.6 p < 0.001). Notably, there was no main effect of ROI suggesting that these results were not driven by differences in the overall mean signal intensity.In order to confirm the statistical significance of the difference in ROI activation across the four conditions, extracted data were compared using repeated measures ANOVA with factors of ROI (6 excluding MOFC), Hemisphere  and Switch Condition . There was a large main effect of ROI  = 6.643 p = 0.001) and importantly, an interaction between ROI and switching condition  = 13.85 p = 0.006). This interaction was driven by greater activation in the mid OFC ROI (anterior) relative to the inferior OFC ROI (posterior) in the acquisition switch condition .The question of whether anterior and posterior LOFC ROIs showed significantly different activation profiles across the 4 switch conditions was examined in an additional ANOVA in which the factors were switch condition (4) ROI (2) and hemisphere (2). The results showed a significant main effect of condition , in theThese findings highlight a prominent role for the LOFC at the point of reversal. However, the question remains, of what the LOFC is doing at the process level during a reversal? One possibility is that the LOFC is involved in affecting change at the point of reversal by overriding the previously rewarded and routine response . An alteThe sensitivity of the LOFC and LPFC to conditions that require changes in behaviour, contrasts strongly with the response that is observed in the medial OFC, a result that accords well with the mounting evidence for a functional divide between the medial and lateral OFC . SpecifiIt has previously been reported that both the ACC and the CN play a role in reversal learning and it has been suggested that the ACC and LOFC have a strong functional correlation . ConsistIn conclusion, the present study has confirmed that different sub-regions of the reversal learning network can be dissociated when different stages of the reversal learning process are examined. It logically follows that different regions preferentially support different aspects of executive function with optimal adaptive behaviour emerging from the interactions of this cognitively heterogeneous executive network as a whole.


Many proteins are observed to localize to the poles within bacterial cells. Some bacteria show unipolar localization, yet under different conditions bipolar patterns can emerge. One mechanism for spontaneous polar localization has been shown to involve the combination of protein aggregation and nucleoid occlusion. Whether the different observed patterns represent global energy minima for the cellular system remains to be determined. In this paper we show that for a model consisting only of protein aggregation along with an excluded volume effect due to the DNA polymer, that unipolar patterns are the global energy ground state regardless of protein concentration and DNA density. We extend the model to allow for proteins to be added to the cellular volume at a constant rate and show that bipolar (or multi-foci) patterns emerge as the result of the system being kinetically trapped in a local energy minimum. Lastly we also consider the situation of a growing cell that starts with a pre-existing aggregate at one of the poles and determine conditions under which either unipolar or bipolar patterns can exist at the point when it is ready to divide. This work sheds new interpretations on recently published experimental data and suggests experiments to test whether such a mechanism can drive patterning in bacteria. Bacillus subtilis have been shown to localize to regions where there are changes in curvature Escherichia coli chemotaxis network, to form periodic patterns due to a diffusion and capture mechanismCaulobacter crescentus the scaffolding protein PopZ localizes in a bipolar pattern yet when expressed in E. coli it tends to only localize to one pole Proteins in bacteria are observed to display a wide variety of localization patterns within the cell, from static polar localization to dynamic waves that propagate along the cell’s length. Proteins that interact with the cell membrane such as DivIVA in The above experiments revealed that some aggregating cytoplasmic proteins can be driven to localize due to the presence of the nucleoid. Recent computational work has shown how the types of patterns depend on the concentration of protein and the volume fraction of the bacterial chromosome E.coli show that many display polar localization One particular system where kinetics likely influences patterning is when protein is added to a cell off of an inducible promoter. Examining the spatial patterns of a library of expressed GFP-tagged proteins in The role of kinetics in localization was studied in computational work on cluster formation on the membrane where it was shown how the rate of addition affects the spacing of clusters In this paper we extend prior work R, with the midcell portion having a length L such that the total length of the bacteria is 2R+L. The diameter of the beads making up the polymer is given by σc and we take it to have the value of twice the persistence length of double stranded DNA, 100 nm. For the protein particles, they have a diameter given by σp, which we have chosen to be σp = 0.5 σc for the results in this paper and take it to represent a nucleated seed of aggregating protein. The number of beads making up the polymer is determined by the volume fraction of the chromosome, fc, which we take to be between 8–16% \pf, that we assume ranges from 0–2.5%. The size of the cell is given by R = 4 σc and unless otherwise specified, for fixed cell simulations L = 16 σc giving an aspect ratio of 3.The model that we simulate to study localization of aggregating particles inside a cell is based on the system published in Saberi et al All beads within the confined volume interact via a Lennard-Jones potential with an interaction parameter and potential cut-off that depends on the types of the interacting particles given by,ij is the average diameter of the two interacting particles, ijr the distance between them, and εij is the interactions strength. It has three possible values: εcc for the interaction between beads on the polymer, εpp for the interaction between proteins and εpc for the interaction between polymer and protein. Both the DNA-DNA and protein-protein interactions are taken as attractive potentials with a potential cutoff equal to 2.5 σij and ijV = 0. The protein-polymer interaction is taken as purely repulsive and has a cutoff equal to 1.122462 σij and ijV = 0.25 BTk. We have the polymer-polymer interaction strength set to εcc = 0.5 BTk to favor weak condensation of the nucleoid, though the results below do not depend strongly on this. We also set the repulsive protein-polymer interaction to have εpc = 1 BTk. Beads in the polymer are also tethered with a logarithmic potential that prevents the bonds from stretching beyond a certain length. The details of the tethering potential do not alter results as a spring potential can also be used. The repulsive portion of the Lennard-Jones interaction keeps beads in the polymer from overlapping.In the above, σa, that we have fixed at a = 0.1 σc. The cell boundaries are treated as hard walls and particle moves that would cause them to leave the simulation volume are rejected. Each simulation starts with the polymer generated in a stretched linear configuration in a cell of the same length. We then gradually decrease the length of the cell, relaxing the polymer, until the cell reaches the desired start length. Then protein particles are added to the cell at a fixed rate, every certain number of MC sweeps, given by the parameter addN. Particles are added until the final desired concentration is reached.We simulate the model using the Metropolis algorithm. At every Monte Carlo (MC) sweep, each particle is randomly moved and its move is accepted or rejected based on the change in energy. We take the move to be uniformly distributed within a sphere of radius, In some of the simulations, specific initial patterns of localized proteins were required, and these were generated by adding proteins in the presence of a localizing potential. This localizing potential consists of having a force at a given position pN proteins at the starting volume, oV, the amount of length that is added is given by addN MC steps, and this amount of volume is added to the cell. Since the cell grows by x coordinates  are scaled by the fractional length change. Every time the cell is grown, the number of beads in the polymer is calculated given the polymer’s volume fraction. If this number exceeds the current number of beads in the polymer, a bead is added randomly between one of the polymers links. The existing two beads attached at that link are expanded slightly along the link direction and the new bead positioned at the midway point. The cell grows until it reaches some final aspect ratio, all the while keeping both the concentration of protein and volume fraction of polymer a constant.We also simulate growing cells where the protein concentration is held fixed over the entire duration of the simulation. In order to do this we add to the cylindrical portion of the cell the volume occupied by a single protein every time a new protein is added to the cell. If there are C. crescentusE. coliIn experiments, a variety of localization patterns are observed from the bipolar patterning of PopZ protein in We consider as a possible mechanism for the observed patterning the presence of the bacterial nucleoid interacting with aggregating protein (see Methods). All of the above experiments show that there is some form of attractive interaction between the proteins. The bacterial DNA acts as a region of excluded volume for the growing protein aggregate. There are no interactions with the cell membrane. (We discuss the addition of other interactions and their effect in the Discussion). In this section we examine the binding energy of the protein aggregate for different concentrations and patterns to address what is the lowest energy state of the system.To answer the above question, we put a fixed number of aggregating protein particles into the volume to generate a fixed concentration of protein. We held the aspect ratio of the cell as well as the volume fraction of the DNA fixed for each simulation. In order to study a specific pattern of localization, we applied a weak constant localizing force at either one or several focal points within the cell volume. This force was applied for 50000 MC sweeps and then turned off. The system was then allowed to relax under zero applied force, monitoring the pattern to make sure that it stayed in the desired configuration. Simulations where the pattern dissolved were discarded. Several hundred sample configurations generated from a particular localizing force were generated and the average energy was calculated.ppε, the same result is observed, namely that it is always energetically more favorable to have one single aggregate than two or more. Regarding how the energy varies as a function of aggregates location, we do not see much difference in the binding energy of a single aggregate whether it is at the midcell or one of the poles. As shown in our previous work The binding energy of unipolar and bipolar patterns as a function of the volume fraction of protein is shown in Thus, based on purely energetic grounds, these results would argue that if the system could reach equilibrium, unipolar patterns should be the dominant observed pattern. Yet experimental results clearly show that for certain systems other multi-foci patterns are the most frequent. In the next two sections we show that these multi-foci patterns result because of the system being out-of-equilibrium, that either the rate at which either the concentration or cell grows can trap the system in these higher energy states.We now consider how the rate of adding aggregating proteins to a cell, starting with none, affects the development of localized aggregates within the cell. The type of experiments that we consider these results to be relevant to would be those in which the concentration of protein within a cell would be increasing with time, such as might be expected when expressing proteins off of an inducible promoter. We expect that if the addition rate is slow enough that the system would find its lowest energy conformation, namely a unipolar pattern. In a pure diffusion and capture mechanism, if particles are added to a cell and then diffuse, we estimate that they would get captured by a pre-existing polar aggregate if the number of time steps between additions were greater than addN. There is no cell growth in these simulations, and cells have a fixed aspect ratio of 3 (we consider cell growth in the following section). In In addN = 1/2500 MC steps. It can be seen that regardless of DNA density, at this rate of addition the pattern transitions from diffuse (red) to a short-lived unipolar situation (green), to a final bipolar pattern (blue). In addN = 1/5000 MC), and now it can be seen that at lower DNA densities, the unipolar pattern persists and continues to grow as more proteins are added to the cell. At higher DNA densities, because the polar aggregate is of smaller size its capture length does not extend the entire cell length and so at this rate it is still possible to form a bipolar pattern. So in this situation, at this given rate of adding protein, cells with lower DNA density would be unipolar while those that have a higher DNA density would be predicted to be bipolar.First we show how the resulting patterns evolve as a function of the growing protein concentration at different densities of DNA within the cell. In addN = 1/10000 MC steps), the unipolar pattern is seen to form at a lower protein concentration than if being added at faster rates due to the kinetics of seed formation. Once formed the unipolar pattern grows and dominates at slow rates, while at faster rates bipolar patterning is possible. The distribution of final patterns at the given final protein concentration is shown as a function of rate and DNA density in addN = 1/10000 MC), the unipolar pattern dominates regardless of DNA density. At the fastest rate, as DNA density increases the ‘other’ pattern starts to have some significant contribution showing that the system is getting quenched into a multi-aggregate state that localizes at other locations beside the poles. In addN = 1/5000 MC) whereas at lower or higher interaction strengths the bipolar pattern appears. The behavior at higher interaction strength can be understood since at a fixed rate the stronger interaction strength will favor more nucleation leading to the possibility of forming a bipolar pattern. At lower interactions strengths, it is certainly not driven by more nucleation, but rather at these intermediate rates the system has time to phase separate the aggregating proteins to the poles, allowing the bipolar pattern to eventually form. How this effect depends on DNA density is shown if pp = 1.3 BTk) as one goes from low to higher DNA density cell types/species, we expect to see a transition from predominantly unipolar patterns to bipolar ones. Lastly, we note that making the aggregating proteins of smaller diameter does not change the qualitative behavior described above, just the timing of aggregate formation and similarly, scaling the cell geometry also does not qualitatively change the behavior, as had been seen previously in For the last part of this section we show how the phase space of patterns changes as a function of the interaction strength between the aggregating proteins. In The preceding section was based on cells of a fixed size where the protein concentration increases linearly in time and the cells started with no formed aggregates of any kind. In this section we explore the consequences of a growing cell, similar to the work on cluster formation in the membrane of growing cells The result for growing a cell at a fixed rate with a fixed concentration of protein and DNA is shown in E. coli (which has a DNA volume fraction ∼ 8–10%) involving misfolded protein or PopZ C. crescentus (that have DNA volume fractions ∼ 16–18%), aggregating proteins such as PopZ would form at the new pole forming a bipolar pattern. When the system starts with a pre-existing polar cluster the existence of other patterns is strongly reduced compared to cells that start with no cluster.In the previous section it was shown that at certain rates of addition, it was possible for cells to be predominantly unipolar at lower DNA densities compared to those with higher DNA density that favor bipolar patterns. In In C. crescentus. In such a system it is reasonable to assume that the concentration of protein is constant during cell growth and the system starts with a pre-existing pattern post-division. This lead to the main difference between the fixed cell and growing cell simulations, namely that in the growing cell we start with a pre-existing aggregate at one of the poles. Thus we start with one aggregate that would have been split into a bipolar pattern in the fixed cell simulations under certain conditions. This results in having to have faster rates than in the fixed cell simulation in order to get the capture length smaller than two cell lengths so that a second seed can form at the other pole. Increasing the protein concentration only aids the formation of the seed at the other pole slightly as the initial starting aggregate is now larger giving a longer capture length. These two effects compete with each other in the formation of the new pole’s seed. Just as in the fixed cell simulation, there were rates at which the capture length was such that a second seed could form and a new foci form at the opposite pole. We also showed that for the same rate of addition/growth that depending on the chromosomal volume fraction it was possible to get either unipolar or bipolar patterns, the latter occurring at higher chromosomal fraction. This could potentially explain the difference between expressing PopZ in E. coli compared to C. crescentus, given that the latter has higher chromosomal density and hence shows bipolar patterning whereas E. coli was predominantly unipolar. Experiments that either expand or condense the nucleoid on cells that have defined patterns would be a possible test of such a prediction. Other recent experiments have shown that there is a spatial organization to genes within the bacterial nucleoid, leading to the suggestion that local bursting of proteins may play an important role in patterning and growth In this paper we have examined the kinetics of nucleoid driven localization of aggregating proteins/particles within a bacterial cell. As shown in prior work, polar localization is favored due to the entropic force exerted by the chromosome on the aggregating particles. Thus seeds are favored to form and grow at the poles. In this paper, we have shown that the equilibrium pattern should be a unipolar pattern, with bipolar or multi-foci patterns being meta-stable patterns. Here we have shown that whether a bipolar pattern emerges depends strongly on the rate of addition of particles. Such patterns result due to kinetic effects, where they exist as stable kinetic traps. A diffusion and capture mechanism governs this behavior, such that if the capture length is shorter than the cell length, then a second focus is possible at the other pole. As shown above, the kinetics strongly influences whether the system will be unipolar or bipolar with the concentration of DNA and protein serving to shift the rate at which this transition occurs. Based on our findings we can interpret some of the observed experimental results on different systems. Our results where protein is added to the cell and the concentration grows in time would be akin to expressing off of plasmids. As shown the fraction of the types of patterns seen depend on rate, and experiments show that sometimes unipolar patterns are favored where as other times bipolar. One possibility for the observed unipolar pattern is that the expressed concentration is simply too low to admit the formation of a second seed. Or it could be the result of the rate at which protein is added, being slow enough so that only the unipolar pattern persists. We would predict that a test of the model in such experiments would be to vary the rate at which protein is added to the cell which could be controlled by the promoter as well as the concentration of inducer. Faster rates, generated by higher inducer concentrations should favor bipolar patterns over experiments where the protein concentration grows slower due to lower induction where unipolar patterns would prevail. Our results from the growing cell simulations was done to connect to endogenous systems such as that of PopZ localizing within the bacteria The qualitative dependence of the localization of aggregating protein in the presence of a bacterial polymer does not depend strongly on the specifics of the interactions, except that they be short ranged. Other work has replaced the bacterial nucleoid with a repulsive potential between the two poles The simulations described in this paper were particle based and stochastic in nature, complementary to the work on patterning in bacterial systems using reaction-diffusion approaches An exciting experimental system that would allow for the direct testing of the mechanisms described here is the Mother Machine Figure S1Non-polar patterning in cells. (A) The fraction of patterns that are ‘other’ as a function of chromosome volume fraction and protein interaction strength for a rate addN = 1/1000 MC. Shown are the frequencies of other for the final range of protein volume fractions in each square. (B) Same as in (A) except using addN = 1/2500 MC. (C) Average distribution of particles as a function of position for ‘other’ patterns at different protein concentrations. At low concentrations, the cluster tends to form midcell. At higher concentrations the pattern consists of a midcell cluster with either one or both poles also containing a cluster.(TIFF)Click here for additional data file.


Modification of polymer properties by blending is a common practice in the polymer industry. We report here a study of blends of cyanurate polymers by molecular modelling that shows that the final experimentally determined properties can be predicted from first principles modelling to a good degree of accuracy. There is always a compromise between simulation length, accuracy and speed of prediction. A comparison of simulation times shows that 125ps of molecular dynamics simulation at each temperature provides the optimum compromise for models of this size with current technology. This study opens up the possibility of computer aided design of polymer blends with desired physical and mechanical properties. Experimis work. . N.B. cue.g. a Dell PowerEdge 1950, 2×Quad Core Intel Xeon E5140 2.33 GHz, 8 GB RAM, 500 GB HDD). The Discover module was used for general simulation requirements, such as geometry optimisation and molecular dynamics as well as molecular mechanical analysis to predict values for tensile, bulk modulus, shear modulus, Poisson’s ratio and the Lamé constants. The Amorphous Cell module was used to build amorphous, homogenous 3D cells composed of molecules that were drawn in silico. It also has a number of protocols designed to make greater use of the Discover module, of specific interest is the temperature cycling protocol, which can be used for Tg prediction. All simulations were performed in the bulk state, i.e. without the addition of solvent as there is no added solvent in epoxy resin cure. Potential energies were calculated using the Polymer Consistent Force Field (PCFF) Legacy module within Materials Studio was used to construct an amorphous cell. The amorphous cell contained an average of 4480 atoms. The simulations were also run with cell sizes double, thrice and four times this number. A unit cell of the appropriate volume was packed randomly with the correct number of monomers, either as single monomers as in the neat resin work or in the correct mole fraction in the case of the blends. The cyanate groups were inspected and those within 4.5 Å of each other were bonded manually to form 1,3,5-triazine rings and a minimization of 5000 iterations was carried out to relieve the strain using the Discover minimization module to a convergence of 1000 kcal mol−1 Å−1. A Conjugate Gradient algorithm − Å−1. As all monomers were assumed to be equally coreactive the nature of the final structure depended ultimately on the initial random arrangement of the monomers. Hence it was possible to generate homogenous networks or blocked semi-interpenetrating networks depending on the starting conditions. Bonds were reacted through the faces of the unit cell to neighbouring periodic boxes to simulate an infinite three-dimensional highly cross-linked network. After the final cyclotrimerization the system was submitted to a final minimization of 10,000 iterations. The NPT ensemble with a time step of 1 fs was utilized with the Andersen thermostat g and the degradation onset temperature.The Materials Studio molecular modelling suite (Accelrys Inc.) was utilised in this work g value more difficult to determine. In essence we are providing more degrees of freedom for the system with the larger models and thereby increasing the error (by comparison with experiment). The equilibration problem increases with larger model sizes. A survey of the data produced by varying cell size is given in Increasing the model size had the effect of improving the statistics of the runs but made the TThe simulations that were run for 1 ns showed that, the statistics (in terms of the scatter of points) were much improved. However, the determination of the Tg value was more problematic, with a shift in the predicted value compared to the experiment. Additionally, the longer timescale served to remove the high temperature transition, which we have assigned to the thermal degradation of the material.g value determined can vary by 30 degrees. This hysteresis is a function of the equilibration problem mentioned above and the cooling down is thought to serve to freeze out high energy motions thereby providing a better correlation with experiment.It has been noticed in previous work by ourselves and others that in order to get results that correlate with experiment, it is necessary to run the simulation by cooling down from the high temperature regime to the lower. If the simulation is run by heating up the system, the TThe plot of simulated density versus temperature from the MD simulation for the neat polycyanurate [BADCy] is shown in g) at 360°C, with the Degradation (Td) at ca. 400°C and 480°C. The higher glass transition temperature results from the much reduced flexibility of this polymer with both of the pendant groups effectively reduced to hydrogens.The simulation of the second neat polycyanurate [LECy]  shows a In previous work we reparameterised the Dreiding force field g) at 295°C and the β transition at 220°C. Interestingly the blending has pushed the degradation onset to 390°C with the second stage at 440°C, which is more similar to the third monomer.The corresponding data for the binary blend (50% of the first two polycyanurates) comprises elements of both networks: displaying a flatter response at lower temperatures Click here for additional data file.Table S2A visual comparison of the experimental TGA and DMTA data plotted against the simulated data.(DOCX)Click here for additional data file.


Fracture subluxations at the proximal interphalangeal joint can be difficult to treat and variable in their outcome.The sling attachment shown was devised in our unit. It is quick and simple to apply to the frame. It is manufactured from Velcro® and Orfit thermoplastic , and is easy to remove for exercise (if appropriate). It does not affect the normal functioning of the frame.Warmed Orfit thermoplastic is bonded onto ‘loop’ Velcro® approximately 2cm from one end.The Velcro® strip is secured around one of the distal portions of the spring at the level of the distal phalanx .The adhesive backed ‘hook’ Velcro® is adhered to the thermoplastic, passed under the distal phalanx and wrapped around the spring on the other side of the Ligamentotaxor® device, fastening to the hook Velcro® underneath, thereby supporting the distal phalanx .Prompt recognition of extensor lag and treatment using this frame modification arrests progression of the problem and facilitates its resolution. We recommend use of the Ligamentotaxor® sling in all cases complicated by extension lag.


In this study we investigate the accuracy of industrial six axes robots employed for patient positioning at the Heidelberg Ion Beam Therapy Center.In total 1018 patient setups were monitored with a laser tracker and subsequently analyzed. The measurements were performed in the two rooms with a fixed horizontal beam line. Both, the 3d translational errors and the rotational errors around the three table axes were determined.For the first room the 3d error was smaller than 0.72 mm in 95 percent of all setups. The standard deviation of the rotational errors was at most 0.026° for all axes. For the second room Siemens implemented an improved approach strategy to the final couch positions. The 95 percent quantile of the 3d error could in this room be reduced to 0.53 mm; the standard deviation of the rotational errors was also at most 0.026°.Robots are very flexible tools for patient positioning in six degrees of freedom. This study proved that the robots are able to achieve clinically acceptable accuracy in real patient setups, too. With the increasing dose conformity achieved in modern radiation therapy techniques the requirements on the accuracy of patient positioning are becoming tighter as well. This is especially true for ion beam therapy where in general the dose gradients are steeper and the dose deposition has a stronger dependency on the materials traversed than in conventional radiation therapy with photons -4). Ther. Ther4])Robots are known for their high reproducibility, i.e. they are able to reach the same position with high precision under the same load. In this study we investigate whether the industrial six axes robots used at the Heidelberg Ion Beam Therapy Center (HIT) are also able to execute the movements precisely with patients having different weights and target points in different locations of the treatment volume. The robots’ movements where monitored during therapy in real patient setups with an external laser tracker. The results presented here can therefore serve in clinical practice as an input for calculating the required safety margins to the clinical target volumes.For the ion beam therapy center in Heidelberg Siemens Medical Solutions adapted floor mounted KUKA robots  to carry the treatment couch  and lateral axis (pitch) up to 15° are in principle possible but are confined in the present version to 5°. The range of possible isocentric angles, i.e. angles resulting from rotations around the vertical axis, amounts to 200°. The patient table is in line with the beam axis at an angle of 270°; from there isocentric rotations of +/- 100° (to 170° and 10°) are possible.The treatment volume, which is the sum of all possible target points, is composed of two cuboids Figure . The craIn order to monitor the movements of the couch we used a laser tracker – an optical measuring device, which is able to measure the position of a retroreflector (target) with a precision of typically a few tens of micrometers . The laDuring the treatment sessions a coarse grid of thin but robust steel wires covered the hole in the floor to protect personnel from tripping into it. This grid, however, hides some of the targets for certain table positions. As long as in total at least three targets and at least one on the caudal and one on the cranial side were detected by the tracker, the position of the couch could still be determined.At the time of the treatment planning CT, skin marks or marks on the patient’s mask are drawn to indicate the position of the reference point. This point is in general not equivalent to the target point, which is determined in the treatment planning system and is typically set in the middle of the target volume. For our (isocentric) treatment technique the target point has to be at the isocenter for irradiation.At the first treatment session the patient marks are aligned with the room lasers. The table coordinates of this position are stored for all future sessions. Then the known offset to the target point is applied.Position verification is performed with a C-arm X-ray imaging system , which is carried by a ceiling mounted robot. For imaging the robot brings the C-arm down from the parking position at the ceiling to the imaging position at isocenter. The X-ray robot can perform full 360° rotation around the table’s longitudinal axis at isocenter height for almost all table angles. The patient positioning is assessed by two, in general orthogonal, images. In most cases this is done at an isocentric table angle of 270°  calculated from the treatment planning CT are matched to the planar images. The algorithm finds that transformation  of the treatment planning CT that minimizes the difference between the calculated DRRs and the X-ray images. The match between the DRRs and the X-ray images is visually checked by the radiation therapists and possible errors are eliminated by manually applied corrections to the registration. The so determined setup correction in all six degrees of freedom is applied together with the isocentric table rotation to the first treatment position.For the patients in this study the required dose was applied by one to three fields at different table angles. Patients treated for head tumors, all of them immobilized with a mask, were tyThe measurements were performed with a FARO laser tracker, model X . FARO provided a software development kit which enabled us to write our own program for controlling and reading out the tracker during patient treatment. A graphical user interface (GUI) implemented in MATLAB allowed the input of all necessary data like table coordinates and patient names.Assuming that the relation between target and table coordinates has been established for one table position, the nominal positions of the targets can be calculated for all other table coordinates. The nominal values serve as a starting point for the search of the actual positions of the targets. Differences between the actual and nominal coordinates can be of the order of one millimeter, depending mainly on the patient load.The measurements can be started from the GUI. As soon as the tracker finds a target, the target coordinates are recorded and the search for the next target starts. In case that the tracker cannot find a target – e.g. because it is hidden behind one of the wires of the grid – the tracker aborts the search for this target and continues with the next. The search for all six targets takes typically 30 seconds, which is far less than the irradiation time of one field.The radiation therapists were asked to perform a measurement in both the verification position, i.e. the position in which the imaging was performed, and in each treatment position. Since no data exchange between the patient record and verify system and the software for the laser tracker measurements is possible, patient names and table coordinates had to be typed in by the therapists manually. The table coordinates are needed to calculate the nominal positions of the targets and hence the starting point of the laser tracker’s search, the patient names for assigning the measurements accordingly. When a measurement is completed the software records furthermore the measured target coordinates and the measurement time. All manually typed coordinates were in a later step compared to the values from the treatment records as stored by the patient record and verify system and corrected if necessary. The correction of the manual inputs is important to exclude that possible typos could fake inaccuracies of the robots.In order to determine the performed table movements both the measured coordinates of the targets in the verification and in the treatment position are required. The actually performed table movements can be calculated by finding the transformation between the measured target coordinates for the two table positions. This transformation consists of rotations and translations in the table coordinate system. The laser tracker, however, measures the target positions in the fixed room coordinate system (IEC fixed). So also the sought transformation between the target coordinates have to be calculated in the fixed system.axis about an axis in the direction of a unit vectorIn general a rotation by an angle αis given by the rotation matrixiso the unit vector is in our coordinate system conventionFor the isocentric rotation by an angle αThe rotation axis of the pitch for the isocentrically rotated system is given byPV is the isocentric angle where the position verification images were taken and αiso the rotation angle between position verification and treatment position.where αFinally the unit vector for roll is given byPV and the setup corrected treatment position Ttr is given byHence the transformation between the target coordinates in the position verification position Twith v being the translational setup correction vector in the room coordinate system.iso, αpitch, αroll and the three components of the vector v) are found by fitting the measured target coordinates of the treatment position to the rotated and translated target coordinates of the verification position calculated with the formula above. The result of the fitting gives the executed movement of the robot. In a final step the executed table movements are compared with the intended table movements and the error in each of the six components is determined.The six parameters defining the transformation  performed from June to September 2010 and 506 in the second horizontal treatment room (H2) from October 2010 to December 2010. The robot hardware is the same for both rooms but for H2 Siemens implemented a different approach strategy to the target position, which avoided inaccuracies due to the backlash of the axes. The cumulated effect of the backlash for all six axes on the target position may amount to several hundred microns. In order to eliminate this effect, each robot movement to any target position is followed by an incremental, axis specific back and forth movement, which ensures that all axis specific movements are always in the same direction at the end of the move. The incremental movement has to be greater than the backlash of the axes. In our case the order of magnitude is well below 1° and the resulting movement of the table top is hardly recognizable by the patient.Figure Since even larger setup errors along the beam direction hardly alter the dose distribution, the error orthogonal to the beam direction is of more importance than the 3d error. This error, we call it the 2d error, has its median value for H1 at 0.25 mm and 0.20 mm for H2. The 95 percent quantile amounts to 0.50 mm and 0.43 mm for H1 and H2, respectively. In Figure It is also instructive to display the error as a function of the isocentric treatment angle Figure . For mosThe histograms for the rotational errors around the three table axis are shown in Figure Robots like the ones installed at HIT are very flexible tools for patient positioning in modern radiation therapy. They allow setup corrections in six degrees of freedom with a resolution of 0.1 mm and 0.1°. In this study we showed that the robots in spite of their high complexity perform their movements also with high precision. This was especially true after the new approach strategy to the final setup position was implemented.In general the setups were performed more accurately when the verification was performed at a table angle close to the treatment angle. But since the accuracy of the analysed movements is comparable to the accuracy with which the X-ray position verification device can be calibrated, setup verification in treatment position does in general not increase the setup accuracy.One has to admit that the initial calibration of the robots is a lengthy procedure. Due to different deflections under different loads the actual positions deviate from the nominal positions slightly. Therefore each robot has to be calibrated by first measuring this deviation for a subset of representative positions (in the order of some hundreds) and different loads ranging from 20 kg to 200 kg. In order to determine the correction for a specific load and position a model is applied, which inter- and extrapolates the deviations between the measured positions and loads.In principle this calibration procedure can be avoided if an external tracking system is used not only for monitoring but also for controlling the movements of the robots. In this study it could be demonstrated that a commercial laser tracker would be a suitable tool to control patient setup in high precision radiation therapy. On the other hand it could also be shown that with a well calibrated system clinically acceptable accuracy can be achieved, too.The authors declare that they have no competing interests.All authors contributed to the design of the measurements and discussed the results. ON, MW and PH performed the setup. ON and MW performed the data analysis. ON drafted the manuscript. All authors read and approved the manuscript.