We study local calcium dynamics leading to a vesicle fusion in a stochastic, and spatially explicit, biophysical model of the CA3-CA1 presynaptic bouton. The kinetic model for vesicle release has two calcium sensors, a sensor for fast synchronous release that lasts a few tens of milliseconds and a separate sensor for slow asynchronous release that lasts a few hundred milliseconds. A wide range of data can be accounted for consistently only when a refractory period lasting a few milliseconds between releases is included. The inclusion of a second sensor for asynchronous release with a slow unbinding site, and thereby a long memory, affects short-term plasticity by facilitating release. Our simulations also reveal a third time scale of vesicle release that is correlated with the stimulus and is distinct from the fast and the slow releases. In these detailed Monte Carlo simulations all three time scales of vesicle release are insensitive to the spatial details of the synaptic ultrastructure. Furthermore, our simulations allow us to identify features of synaptic transmission that are universal and those that are modulated by structure. Chemical synaptic transmission in neurons takes place when a neurotransmitter released from a nerve terminal of the presynaptic neuron signals to the postsynaptic neuron that an event has occurred. The goal of our research was to model the release at a type of synapse found in the hippocampus, a part of the brain that is involved with learning and memory. The synapse model was simulated in a computer that kept track of all of the important molecules in the nerve terminal. The model led to a better understanding of the extant experimental data including exact conditions that lead to the release of a single packet of neurotransmitter. According to our model, the release of more than one packet can be triggered by a single presynaptic event but the packets are released one at a time. Furthermore, we uncovered the mechanisms underlying an extremely fast form of release that had not been previously studied. The model made predictions for other properties of the synapse that can be tested experimentally. A better understanding of how the normal synapses in the hippocampus work will help us to better understand what goes wrong with synapses in mental disorders such as depression and schizophrenia. Exactly why this occurs has not yet been understood.Because of its small size and lack of active zone redundancy, hippocampal vesicular release is a highly stochastic process. The most basic feature is the release time course, in units of probability of release per unit time, after a single depolarization. Measurements of this time course have revealed several interesting features. First, the data reveal the existence of multiple time scales involved in this release Furthermore, some experiments have found evidence of a short refractory time (∼ 5–7 msec) after single vesicle release, a time during which additional release is precluded Here, we construct a stochastic spatially-explicit computational model that enables us to realistically simulate the intracellular calcium dynamics in the presynaptic bouton, tracking in detail the progression from depolarization to vesicle release. To do this, we will rely on known ultrastructural details of the CA3-CA1 synapse and also on recent ideas regarding calcium sensor proteins that control the release machinery. The model can be validated with existing release time course data and will be used to address the issues sketched above. We will also consider the effects of genetically knocking out parts of the calcium sensor. Future work will discuss how structural information regarding synaptic geometry and synaptic components can be inferred by combining this model with new measurements.As will be seen below, our model leads to several important findings. First, we show that in general vesicle release occurs with three distinct timescales. Aside from the fastest one, which is directly controlled by the calcium profile, the other timescales are determined by the sensor kinetics and hence are almost independent of the detailed synaptic geometry. Next, we show that the aforementioned notion of a refractory period is necessary for explaining release data at high probability synapses. Finally, we demonstrate the role of asynchronous release in modulating short-term plasticity. These results help make sense of existing disparate data as well as offer specific predictions for future experiments on hippocampal synapses.Exocytosis, the process by which vesicles bind to the membrane and release their neurotransmitter cargo, is primarily triggered by the VDCC calcium currents. The arrival of an axonal action potential  from the VDCC cluster (source of Ca2+ flux) 2/s, close to experimentally measured values 2/s in the cytoplasm) and our local calcium profiles compare well with those of other studies The canonical CA3-CA1 en passant synapse geometry used in our simulations is shown in Our basic protocol is to simulate the sequence of events at the CA3-CA1 synapse beginning with the arrival of an action potential, the opening of the VDCC's, the diffusion of calcium from the VDCC's to the calcium sensor and the triggering of vesicle fusion and glutamate release Release at a single active zone with seven docked vesicles is governed by a dual calcium sensor kinetic scheme . The duaIn our baseline model, simultaneous release of multiple vesicles is prevented by imposing a refractory period of 6 ms after a release event takes place cl ) which only effect stimulus response. We have verified that the spontaneous release rate in our model  that rarely ever release to synapses with high release rates (rp>0.9) cl) and the number of VDCC's. cl). The number of VDCCs present in the cytoplasmic membrane regulates the calcium flux at the specified cl. Small cl leads to sharper, narrower local calcium peaks at the active zone  and tslow (140.0± 28.0 ms) in agreement with the reported data fastt). Remarkably, we have been able to accomplish this fit without having to alter the binding affinity of the synchronous pathway, which remains at 38 µM. This affinity is the primary determinant of the calcium sensitivity, since the fast component contributes more than 90% to the overall release probability . The neurotransmitter release peaks after a typical latency of ∼3 ms. Note that here we measure the latency starting from the beginning of the action potential  . This result of the model is consistent with reported data from high and low release probability synapses that show similar decay constants As mentioned above, the model synapse achieves rp = 0.2, the likelihood that more than two vesicles are released was less than 5%. Furthermore, there is less than 20% chance of releasing more than 2 and almost never more than 6 vesicles for rp = 0.6 and a 33% chance of releasing more than 2, and almost never more than 9 vesicles for rp = 0.95. The size of readily release pool (RRP) has been estimated to be 5–10 vesicles at CA3-CA1 synapses As mentioned above, our model posits that multiple releases can take place from the active zone after a refractory time constant of ∼6 ms following each release Stevens and collaborators introduced the idea that there is a short refractory time following vesicle release from an active zone. With such a refractory period more than one quantum of neurotransmitter can be released by an action potential, but the quanta are released one at a time. Several recent experimental studies have tried to address the question of refractoriness after release but with conflicting results. Explicit measurements at a wide variety of synapses conclude that there exists a “one active zone-one vesicle release” principle and hence provide direct evidence for functional coupling within the active zone rp  = 0.2, the release transient for a synapse with a refractory period (gray line) is almost indistinguishable from a synapse without any refractoriness (black line). Thus for this set of parameters, the presence or absence of refractoriness does not make any functional difference. For a release probability of rp = 0.2 for the whole active zone, each of the 7 individual docked vesicles must have a release probability of 0.031 so the probability that 2 or more vesicles being released is only 0.02. This implies that although any single vesicle was released on 20% of the stimuli, two or more vesicles were released on only 2% of the trials. The detailed timing of release of the second vesicle relative to the refractory period has a negligible effect on the overall averaged release profile. The consequence of a refractory period was more prominent for rp = 0.95. For a synapse with independent releases (i.e. no refractory period) and rp = 0.95, 2 or more vesicles were released on 67% of the trials. The top panel in Our basic strategy is to compare neurotransmitter release profiles with and without the existence of a 6 ms refractory time constant preventing simultaneous release of different vesicles. We do this comparison for different values of the overall release probability See . For a rWe have seen that our model can reproduce one of the important distinguishing characteristics of neurotransmitter release in hippocampal CA3-CA1 synapses, that the decay time scales are conserved across a wide range of release probabilities even as the overall amplitude of the transient is modulated rp  = 0.95 , the calcium sensor for neurotransmitter release, eliminates the fast release component of the transient but leaves the slow component intact We can understand this effect in more detail by focusing on the change in time-course brought about by the genetic modification. For a synapse without refractoriness, the ratio  between Refractoriness differentially affects synchronous and asynchronous release at early and late times after a single stimulus and this effect is sensitive to the initial release probability . But whar  = 0.2 , pr  = 0.6 , and pr  = 0.95  is shown in r  = 0.6 the facilitation (ratio of first two release rates) in the synapse with refractoriness (black line) was almost twice that of a synapse without refractoriness (grey line). However for the synapse with refractoriness the background release level (due to asynchronous release) was much higher compared to a synapse without refractoriness. These predictions can be directly tested in future hippocampal synapse experiments.The response of our model synapse for the different cases of initial release probabilities pWe now wish to investigate the role of the slow sensor in the presence of a spike train, the response to a 10 Hz stimuli for a total of 400 ms (i.e. 4 triggers) for a synapse with intrinsic release probability 0.2 is shown in aγ, the fusion rate of vesicles activated by the asynchronous sensor, is smaller than the corresponding rate for the synchronous one. To demonstrate why this is necessary, a sample release profile of the asynchronous pathway for our single active zone synapse with 7 docked vesicles rp = 0.2, number of VDCC = 48, lc = 250 nm). The early peak in this figure, present for simulations at all values of the release probability, is clearly inconsistent with electrophysiological data All the results given so far have used a model for which the parameter γ needs to be significantly slower for the asynchronous pathway relative to the synchronous pathway. This introduces an additional parameter ‘a’ such that the neurotransmitter fusion rate is a  = aγγ (with a<1) for asynchronous release (see a = 0.025 (i.e. net asynchronous vesicle fusion rate = 50/s), the early release from the asynchronous pathway is suppressed and all the detailed characteristics of neurotransmitter release can be reproduced  and multiple release events may occur, subject to the refractory time constant. With this framework, it is possible to consistently reproduce all our data, including the 3 timescales and a cumulative release well matched to the RRP (data not shown). In short, an additional parameter ‘a’ is needed in the docked vesicle model with individual sensors on each vesicle, to directly suppress asynchronous release, whereas in an alternative phenomenological approach that treats the whole active zone as having a single gating mechanism, no such parameter is needed. We have chosen to focus on the individual vesicle model, as there is no obvious justification for such a strong vesicle coupling.In the context of a model with Neurotransmitter release at chemical synapses in response to electrical stimulus is tightly regulated over multiple time scales by mechanisms in the presynaptic terminal. Release takes place at specialized locations at the presynaptic membrane called active zones designated by the presence of SM (Sec1/Munc18-like) proteins We have chosen to carry out our study focusing on the CA3-CA1 synapse in the hippocampus. The advantage of using this synapse is its relative simplicity, consisting of only one or two active zones, and its starring role in many studies of plasticity. Even with this emphasis, varying results from different experiments have led to confusion regarding certain basic features of synaptic transmission. Our computational experiments have led to possible resolutions for some of these contentious issues, such as the existence of refractoriness between releases, cohesively bring together data from different sources that point to universal features of vesicle release and those that may be unique to the CA3-CA1 synapse 2+ channels and relative location of Ca2+ do not modulate the relative dynamics of asynchrony to phasic release. This study strongly supports our own modeling results in which the calcium sensor governs all the relevant time scales. This result stands in contrast with other approaches In particular, our simulations have illuminated the observation in two separate sets of data fastτ and slow,τ) or (superfastτ and fastτ) have been reported Two decay timescales have indeed been observed in hippocampal synapses. Also, similar findings (a slow decay component of ∼82 ms) have been reported in parvalbumin-containing GABAergic interneurons expressing P/Q calcium channels As has been explained, our model for the calcium sensor is a modified stochastic version of the one introduced by Sun et al. A detailed comparison of our model for vesicle release and that of Sun et al. is outlined as follows. In contrast to the deterministic kinetic sensor model of Sun et al., our model is a spatially explicit stochastic model of the entire bouton. In Sun et al. 2+ sensor with a long memory as described in our hippocampal model can have a significant role in activity-dependent short-term synaptic plasticity  of the entire active zone where the spurious early peak in asynchronous release is eliminated without having to change the vesicle fusion rates. The overall effect of this inhibitory coupling is to reduce the effective asynchronous neurotransmitter fusion rate. Developing this possibility further would require a better understanding of the proteins that are responsible for the coupling and including the concomitant explicit sensor-sensor coupling in the kinetic scheme. Experimentally, one would need to develop knock-outs of the coupling proteins and test these for evidence of enhanced asynchronous release rates, especially the existence of an early release peak not present in wild-type synapses.However, it is only possible to fit all the release data for CA3-CA1 synapses when we set the value of the neurotransmitter fusion rate, Finally, our study is built upon an underlying assumption that spontaneous release, synchronous release and asynchronous release take place from the same RRP Simulations were performed using MCell, version 3 2/Sec) 2/sec specified for the simulation and arises because our model has an accurate description of the calcium binding kinetics of mobile calcium binding proteins in the synapse (See b (tb  = 10 ms or 1 ms ). The value of ‘r’ in every bin can be determined by r = release rate . N . tb . The docked vesicles were clustered in a hexagonal array with largest center-to-center distance between vesicles of 35 nm.The apparent diffusion constant of calcium, a key parameter for physiological relevance of our results, was matched in the model to the measured value (50 µmText S1Supporting information.(0.23 MB PDF)Click here for additional data file.


Based on the assumption that in a standard eco-dumping model governments are uncertain about future product demand and allowing governments to obtain information from firms, we examine governments’ and firms’ incentives to share information. We show that when governments regulate polluting firms through emission standards, then governments and firms will reach an agreement concerning information sharing. The opposite holds when governments regulate pollution through emission taxes. In their seminal study Brander and Spencer } + (1 + g)2 (3 + g)2 k (since the problem is concave we neglect the second order conditions). δz/δZ < 0 implies that domestic and foreign emission standards are strategic substitutes  + (1 + g)(3 + g)2 k and the superscript (cc) indicates that governments and firms in both countries share information. These are the equilibrium levels of outputs and standards. In case the demand is high (θ is high) then both outputs and standards are high and vice-versa. To determine profits and welfare we substitute equilibrium values given in (9) in (3) and (4) respectively :Solving simultaneously Stage 4 equilibrium output (7), the domestic government’s reaction function (8) and the corresponding equations for the foreign firm and government we obtain the Subgame Perfect Nash equilibrium:ex ante profits (10) and ex ante welfare (11) depend positively on var(θ). This is due to the convexity of the profit function with respect to the demand intercept. Hence, as the variability of θ increases the expected values of profits and welfare also increase.In Stage 1 of the game from the governments perspective θ is unknown and thus the expected profits and welfare are:3.2.Now, we examine the scenario where the firms and the governments do not share information. If this is the case then the governments act under incomplete information as θ is unobservable for them. Thus, the equilibrium notion that we use is Bayes Nash equilibrium. Firms’ maximizing problem follows in the lines of the previous analysis, while welfare maximization follows a slightly moderated one. Since θ is unobservable to the governments, yet they know the distribution that it follows, they maximize their expected welfare with respect to the emission standard. This results to an equivalent reaction function given in (8) after setting θ = 0.i.e., increase z, compared to the first best case where regulation is set such that the marginal cost of abatement and the marginal damage from pollution are equated, i.e., δπ/δz = δd/δz, thus the externality is fully internalized and then, the strategic effect is zero. In order to determine the level of expected profits and welfare in the case of standards we substitute the equilibrium values given in (12) and the implied abatement level by (1), into (3) and (4) respectively. Taking expectations and after some algebraic manipulation we get:Solving simultaneously (7) and (8) after setting θ = 0 as well as the corresponding equations for the foreign firm and government we obtain the Bayes Nash equilibrium:i.e., ex ante profits and welfare increase with uncertainty. This is true because the firms select outputs after θ is revealed. From a technical aspect two opposing effects determine this outcome. A positive effect is due to the convexity of the profit function in terms of the demand intercept and it is similar to the ones introduced by Cooper and Riezman  > E[ncw], E[ccπ] > E[ncπ]}and {E[cnw] > E[nnw], E[cnπ] > E[nnπ]}.Proof in AppendixUsing Lemma 1, we define the Nash equilibrium of the information sharing game in the following proposition:Proposition 1 When emission standards are the policy instrument in use, then with unknown common demand:vice-versa. At the same time when regulation is laxer the firm is more aggressive in international competition gaining a larger market share and higher rent shifting from the rival firm. Hence, the mechanism which supports laxer regulation as a commitment device over the rival firm is re-enforced benefiting both the government and the firm. Although at times of lower demand the opposite holds, the losses suffered in this case are lower than the gains at times of higher demand. Therefore, adjusting standards to demand makes the welfare and the profit functions more convex and in turn increase their levels. It is also important to note that the validity of this argument does not depend on whether the rival pair shares information or not. The expected levels of profits and welfare always depend positively on the variance of the demand intercept.Proposition 1 states that as it is a dominant strategy for the governments and the firms to share information it is also a Nash equilibrium of the game. The benefits from sharing information are greater than the losses. In particular, the benefits for the firms and the governments from sharing information arise from the convexity of the profit functions with respect to the demand intercept. The losses are attributed to the convexity of the damage function of pollution with respect also to the demand intercept. When the firms decide to share their information with the governments then in exchange they get laxer regulation (higher standards) as standards depend positively on θ see  in good Given this result which so far parallels the one of CM, although in a different context, it is interesting to check if the result of sharing information is socially desirable. This is true when the expected welfare level in the sharing information case is higher compared to the case where none of the two pairs share information. Proposition 2 summarizes this comparison:Proposition 2 When emission standards are the policy instrument in use, then with unknown common demand:Expected welfare under information sharing is higher compared to the corresponding one where the governments do not receive information, i.e., E[ccw] > E[nnw].Proof in AppendixThis result is of major importance since it states that when emission standards are the unique policy instrument in use, information sharing occurs and this is superior in terms of expected welfare compared to the case where the two pairs do not reach an agreement. Put it differently, from the social perspective the Nash equilibrium is socially optimal. At the same time it can be shown that each firm and government prefer that the rival pair do not share information regardless of their agreement (the proof of this claim is neglected for brevity and it can be provided upon request by the authors). If the domestic players share information, then the domestic firm and government are better off if the rival pair do not reach an agreement. In this scenario, the domestic firm faces more flexible standards which in turn, when demand is high, allow the domestic firm to obtain an even larger market share, while in the opposite case the market share shrinks. The benefits attributed to the convexity of the profit function with respect to the demand intercept are now higher. If the domestic pair do not share information they prefer that the rival pair does the same. If not, then at times of high demand the rival government indirectly subsidizes the corresponding firm through laxer regulation shrinking the market share of the domestic firm and reducing so its expected profits. Contrary to that, when demand is lower then regulation is stricter benefiting the domestic pair who decided to not share information. However, the first outcome prevails to the second one. The fact that each pair prefers that the rival pair is not informed does not lead to an “informational prisoner’s dilemma” as CM claim in their model. Put it differently, when emission standards are used to subsidize exports instead of subsidies, information sharing leads to a superior outcome in terms of expected welfare. It is interesting that even if both pairs prefer that the rival one does not, indeed they do share information and this is mutually beneficial compared to the case where they do not. The benefits arising from the convexity of the profit function with respect to the demand intercept when the two pairs reach an agreement outweigh the expected welfare losses attributed to the variability of standards and, thus, the variability in the damage from pollution.4.4.1.In contrast to the previous case we now assume that both governments use taxes to control pollution. Now firms have two control variables available, output and the abatement level. Solving backwards we derive the first order conditions for the domestic firm:i.e., marginal cost plus marginal cost of abatement) due to the existence of the maximum standard constraint, while implementation of a tax implies horizontal total marginal cost of production  as it increases proportionally to the tax. It follows that given a change in the output decisions of the rival firm, the responsiveness of the home firm’s output is greater when a tax rather than a standard is implemented. The profit maximizing condition with respect to abatement is given by a does not appear into the profit function. The equilibrium values of outputs as a function of taxes are obtained by solving the domestic and foreign firms’ reaction functions simultaneously:The output reaction function of the domestic firm is implied by τ1 = g(9 + 4g) + (3 + 2g)2 k. If 3k + g(−1 + 2k) > 0 then the reaction function of the domestic regulator implies that taxes are strategic complements, which as we will see later is a sufficient condition for the existence of an interior solution in equilibrium, otherwise we obtain a negative pollution tax.Examining the domestic government’s decision about the optimal tax we maximize welfare with respect to the emissions tax. Thus, for home we have:τ2 = g(9 + 5g) + (3 + g)(3 + 2g)k and the subscripts (t) and (T) denote that taxes are implemented as a policy instrument. As already mentioned, In order to obtain the equilibrium levels of outputs, taxes and pollution in the two countries we solve simultaneously Substituting the new equilibrium levels in As in the case of standards, we the ex ante values for profits and welfare as follows:From 4.2.In case that the governments and the firms do not reach an agreement the governments act under incomplete information. Firms’ maximizing problem remains the same as in the complete information case, while welfare maximization is moderated. Now, the two governments maximize their expected welfare with respect to the emission standard. Welfare maximization by the domestic government yields a reaction function given in (18) after setting θ = 0.Solving simultaneously (1), (16), (17) and (18) after setting θ = 0 as well as the corresponding equations for the foreign firm and government we obtain the Bayes Nash equilibrium:Comparing the solutions given in (22) and (19), several inferences can be drawn that play a significant role in determining the expected national welfare levels. When governments are not informed, the level of emission taxes in each country is determined at a specific level and it is not affected by θ. Contrary to the situation where the governments and the firms share information the governments do not adjust their policy to θ and, thus, when θ is positive the firm may adjust its output without being penalized by the government. This, together with the fact that now abatement does not depend on θ, creates a clear disincentive to the firms to reveal their private information.However, this is not true for the governments. If we compare the level of pollution in equilibrium in the two polar cases we obtain that pollution is higher in the incomplete information case when θ is positive and lower if θ has the opposite sign  and the implied abatement level in (16), into (3) and (4) respectively. Taking expectations and after some algebraic manipulation we get:It is clear from (23) that expected profits depend positively on var(θ). From 4.3.To complete the full payoff matrix, the asymmetric cases and the expected values of profits and welfare must be calculated see . Given tLemma 2 When emission taxes are the policy instrument in use, then with unknown common demand:It is a dominant strategy for the government to accept information and for the firm to not share information regardless of what the rival pair does, i.e., {and {Proof in AppendixUsing Lemma 2, we define the Nash equilibrium of the information sharing game in the following proposition:Proposition 3 When emission taxes are the policy instrument in use, then with unknown common demand:Proposition 3 states exactly the opposite of Proposition 1. Now, the governments and the firms do not reach an agreement since the firms are unwilling to reveal their private information about θ. That is because in the case that the firms supply their information to the governments they will adjust the tax accordingly. For example, if θ is positive, then the government raises the emissions tax further, while if θ is negative then the government cut the tax by In order the results of this section to be comparable with those of the previous section we will see if the Nash equilibrium coincides with the optimal solution from the social perspective. Proposition 4 illustrates this comparison:Proposition 4 When emission taxes are the policy instrument in use, then with unknown common demand:Expected welfare under information sharing is higher than expected welfare when the two pairs do not share information, i.e., Proof in AppendixPut it differently, Proposition 4 suggests that the Nash equilibrium with taxes is sub-optimal from the social perspective. The residents in the two countries would be better off if the firms and the governments share information, even though this does not happen in the Nash equilibrium of the game.5.i.e., price or quantity competition, but the mode of competition of the governments is equally important, i.e., price or quantity instrument. In particular, the use of a price instrument daunts the participants from a possible agreement, while regulation through quantity constraints encourages all parties towards an agreement.In this paper we examine the issue of information sharing in a strategic trade model where the exporting firms yield a pollutant as a by-product of production. Environmental policy instruments, emission standards and taxes, instead of the traditional trade instruments are implemented. Similarly to CM, we examine whether the firms and the governments will reach an agreement concerning information sharing. Contrary to CM, our results suggest that when emission taxes are used the firms are unwilling to reveal information. As a result an agreement, although socially desirable, is not achieved as a Nash equilibrium despite the fact that the firms compete in quantities. The main contribution of this study is that not only the mode of competition of the firms matters, The suggestions of this study can be further extended. For instance, the fact that the goods may be consumed in the two exporting countries and thus consumer surplus is a determinant of welfare may affect the decisions of the governments concerning the level of regulation but not the decision to reach an agreement as long as the driving forces of the mechanism remain in place. This is true also if we allow for a higher number of firms or for pollution to be trans-boundary. If this is the case, again standards (taxes) will (not) induce the participants towards an agreement as in good times environmental policy will be laxer (tighter), while in bad times will be tighter (laxer). Therefore, the results are expected to differ only quantitatively since the damage in the case of trans-boundary pollution is expected to be higher. Another modification of the model concerns the mode of uncertainty. Firms might hold private information about their costs of abatement instead of the common demand. Put it differently, the governments do not know the exact level of abatement costs and, thus, they set up an agreement to gain the extra information. We shall expect again that, in the case of standards, contrary to that of taxes, information revelation exploits further the convexity of the profit function, enforcing a bilateral agreement. Even if the functional forms used are generalized such that an interior solution exists we expect that the basic implications of this study will be replicated, since the governments are expected to increase the variability of production through standards when the firms reveal information, while the implementation of taxes reduces the variability in production under complete information driving them to keep private their information.


A small group of patients with manifest Huntington's disease (HD) were followed longitudinally to assess cognitive decline in relation to time from disease diagnosis. This article looks at performance on a range of computerised and pencil and paper cognitive tasks in patients 5 years post diagnosis, who were assessed annually for a 5 year follow up period. The almost universal cognitive decline reported in other longitudinal studies of HD was not replicated in this study. It was proposed that longitudinal follow up in HD is complicated by the varying degree to which different tasks are able to withstand repeated administration; a finding which would have significant implications on study design in future trials of cognitive enhansing interventions. Huntington’s disease (HD) is an inherited progressive neurodegerative disorder that is characterised by a triad of motor, cognitive and psychiatric problems Huntington’s disease is thought to affect 6-7 people per 100 000 HD was traditionally thought to be mainly a movement disorder at least in the early stages and a clinical diagnosis was based solely upon the motor features of the disease. It is now widely accepted that cognitive and psychiatric disturbances are almost universal, begin early in the course of the disease, are heterogeneous and contribute substantially to the functional disability of those affected.  Anecdotal evidence from families and caregivers along with recent findings suggest that it is the cognitive and psychiatric profile of HD that contributes most to the patient’s quality of life and behavioural functioning, and not their motor phenotype Cognitive deficits associated with HD early on in disease are consistent with disruption to the functional integrity of the frontostriatal circuitry Despite the advances in HD research in the last 20 years, there have been very few major break throughs in terms of medical treatments. The majority of drugs currently used in clinical practice are done so not based on reproducible verifiable evidence but more on anecdotal clinical experience. This is most prominently the case for both the cognitive and psychiatric challenges of HD as, to our knowledge, only one placebo controlled study has been carried out to date that focuses on either of these areas The ultimate hope for new and emerging HD therapeutics is to develop a neuroprotective compound capable of slowing down or even stopping the progression of the disease and ultimately preventing the subtle early signs deteriorating into manifest disease. To date no such drugs have survived clinical testing i.e. none have shown that they can truly slow down disease progression. With the absence of any neuroprotective agents there is still a great need for new and more effective symptomatic treatments. While therapeutics designed to treat the cognitive and psychiatric symptoms of HD are undoubtedly the future of HD pharmacological research, we are not yet in a position to achieve this effectively in practice. At present, cognitive performance has only been used as the primary endpoint in one clinical trial where performance was measured using the Mini Mental State Exam (MMSE); a measure of global clinical functioning. In order to see the addition of cognitive and psychiatric endpoints to the protocol of large pharmaceutical studies there is a need to better understand the impairment experienced by patients throughout the course of the disease. Appropriate tools are needed that reliably measure longitudinal change specifically in HD  to stand any hope of demonstrating the efficacy of a cognitive enhancing medication.   Eighteen patients were recruited at the Cambridge Centre for Brain Repair, Cambridge UK between 1995 and 2004. They gave informed consent and were enrolled in an ethically approved study of cognition in HD. Of these 5 patients went on to be given fetal neural transplantation as part of a separate study and four patients withdrew their consent without completing 5 years of follow up. These 9 have been excluded from this analysis. All patients had mild to moderate HD at time of enrollment as measured by the UHDRS For the purpose of this article patients were grouped according to their time from disease diagnosis. Patients were assessed annually for 5 year period starting at 5 years from disease diagnosis, on the following paper and pencil tasks (in order of administration): 1) Verbal (letter) fluency: a test measuring the spontaneous production of phonemically related words 2) Verbal  fluency: a test measuring the spontaneous production of semantically related words.3) MMSE: A test of global cognitive functioning4) Stroop Test: a test of executive function that looks at selective attention and interference susceptibility 5) Symbol Digit Modalities Test: primarily a test of mental speed, motor speed and mental flexibility6)Retain Trail Making test: a timed test of scanning and visuomotor tracking, divided attention, and cognitive flexibility 7) Digit span – forwards and backwards: a test of immediate and delayed verbal recall 8) Rivermead Behavioural Memory Test Battery – the story recall test : a test of immediate and delayed verbal memory.  In addition each patient has completed five consecutive years of data on the Cambridge Neuropsychological Test Automated Battery (CANTAB); the battery consisted of the following tasks: 1) One Touch Stockings of Cambridge – a test of planning and working memory 2) Spatial Working Memory: tests spatial working memory by measuring the ability to search systematically for hidden tokens among several boxes without returning to boxes where tokens had been found previously 3) Matching to Sample Visual Search – a test of attention capable of dissociating visual search speed and motor movement speed.   4) Spatial Span: tests spatial memory span by measuring the ability to memorise the order in which an increasing number of white boxes change colour.  5) Pattern Recognition Memory: ability to recognize visual patterns following a short delay6) Spatial Recognition Memory: measures ability to recognize spatial orientation of visual stimuli following a short delay.  The assessment session took between 1 and 2 hours to complete although this time lengthened as patients became more symptomatic.  In addition to the neuropsychological battery all patients were assessed using the motor UHDRS which was completed by a qualified and experienced neurologist.  To examine longitudinal decline a mean value of all data points was calculated for each time point. A linear regression model was used with time from diagnosis as the independent variable and patient test performance as the dependent variable.  All longitudinal changes reported are significant to at least the p < 0.05 level. For a detailed summary of test results see Table 1 and Table 2. Performance on the verbal fluency task for semantic and phonemic categories declined subtly over the 5 year follow up, as did performance on the interference stage of the Stroop task, the backward span length on the digit span task, total correct on the CANTAB One-Touch Stockings of Cambridge and the total score on the CANTAB Spatial Recognition Memory task.  A more marked decline, as indicated by the steeper regression slope, was seen in the word reading and colour naming stages of the Stroop task, performance on the Symbol Digit Modalities task and the search time on the CANTAB matching to sample task.  The remaining tests were not found to deteriorate over time in relation to time from disease diagnosis. This study examined the longitudinal cognitive performance in a small cohort of symptomatic HD patients over a 5 year period. As expected, this group of patients showed a significant deterioration in motor performance over this time, as did their cognitive performance, but only in some domains.  Previous studies have reported an almost universal deterioration in cognitive tests of attention and executive function in longitudinal studies with shorter follow up periods This lack of consistency could be explained in several ways. Five years is an unusually long time for cognitive follow up and it is likely that prolonged testing, annually, on tasks that remain unchanged, could produce learning effects that may confound the results; i.e. it may be that any deterioration is offset by these learning effects. For example in a task such as the Retain Trail Making Test where the learning is likely to be procedural as apposed to episodic, patients may develop strategies over time that allow them to complete the task more efficiently rather than remembering exact answers. This would allow patient to become artificially good at tasks and render the tasks inept at tapping into any cognitive impairment that may be there. Additional, it is highly likely that cognitive performance does not deteriorate in a linear fashion on all tasks. Unfortunately with such a small sample size it is difficult to ascertain whether this is the case in this group of HD patients.        Getting a clear understanding of cognitive performance over a long period of time in HD is important and potentially vital for future therapeutic trials. While conventional clinical trials tend to only last for 3 years or less, transplantation trials and trials of genetic therapies require longer follow up before being able to conclude the efficacy of such treatments. In addition, clinical studies looking into disease modification in HD will need to be able to demonstrate that the intervention has stopped or slowed the progression of the disease for a significant amount of time. Our findings suggest that obtaining a clearer understanding of cognitive change in HD over a long period of time is complex. While some tasks are useful cross-sectionally they may not be helpful longitudinally. Additionally, tasks which are capable of mapping deterioration over a short period of time may not be so reliable for extended follow up periods. It is important to get a good understanding of how cognitive tasks behave over time to allow us to design the best protocols for future disease modifying pharmaceuticals or surgical interventions.  Funding Information:This work was initially supported by the MRC.Competing interests:The authors have declared that no competing interests exist.


Despite significant progress in imaging technologies, the efficient detection of edges and elongated features in images of intracellular and multicellular structures acquired using light or electron microscopy is a challenging and time consuming task in many laboratories.We present a novel method, based on the discrete curvelet transform, to extract a directional field from the image that indicates the location and direction of the edges. This directional field is then processed using the non-maximal suppression and thresholding steps of the Canny algorithm to trace along the edges and mark them. Optionally, the edges may then be extended along the directions given by the curvelets to provide a more connected edge map. We compare our scheme to the Canny edge detector and an edge detector based on Gabor filters, and show that our scheme performs better in detecting larger, elongated structures possibly composed of several step or ridge edges.The proposed curvelet based edge detection is a novel and competitive approach for imaging problems. We expect that the methodology and the accompanying software will facilitate and improve edge detection in images available using light or electron microscopy. The analysis of microscopy images is a time consuming task in many microbiological and biomedical laboratories. There is an ever-increasing need for analyzing large numbers of images acquired with microscopes in connection with different assays, where one wishes to measure the number of cells, the size of certain objects, the area occupied by cells, etc . In receThe problem of edge detection has a long history in computer vision , x1 = 0, 1,..., N1 - 1, x2 = 0, 1,..., N2 - 1, whose discrete Fourier transform (DFT) isWe introduce the discrete curvelet transform applied to an image with intensity values given by the function f into the curvelet coefficients cjlk, such thatThe discrete curvelet transform is now a decomposition of the image k =  and φjlk is the curvelet on level j with direction l and spatial shift k. Additionally, the curvelet transform preserves l2-norms, i.e.where f into J detail levels, with Lj directions on each level, and Kjl,1 × Kjl,2 spatial shifts for each of these directions. The curvelet φjlk is defined through its discrete Fourier transform asThe discrete curvelet transform thus provides a decomposition of the image andθl. The slopes defined by the angles θl are equispaced. Uj is a frequency window function with compact support, which is approximately 1 inside a wedge around the n1-axis with 2j ≤ n1 < 2j+1, and decreases quickly to zero outside this area. The Uj are defined so thatHere, Thus, the discrete curvelet transform provides a decomposition of the frequency space into dyadic rectangular coronae, each of which is divided into wedges, the number of which doubles with every second level.φjlk. The imaginary part is odd with respect to its center, while the real part is even.In figure cjlk are computed asThe curvelet coefficients φjlk, which in the wrapping version is wrapped by periodicity onto a rectangular area near the origin, namely 0 ≤ n1 <Kjl,1, 0 ≤ n2 <Kjl,2. The size of this rectangle also determines the ranges for the spatial shifts k1 and k2, and therefore the resolution at curvelet level j.This scalar product is computed as a sum over a parallelogram containing the support of Uj, the curvelet profile has the same shape at all scales.The curvelet transform can be seen as an efficient computation of convolutions with filters with profiles as in figure j = 1, the curvelets are non-directional (i.e. L1 = 1) and are similar to the Meyer wavelet scaling function  or wavelets . If curvelets are used on the finest level, they may be included in the edge detection procedure below like any other curvelet level. Throughout this paper, however, we have used wavelets on the finest level, because of the shorter execution time and smaller memory requirements, but also because for our examples we were not interested in directional information on the smallest scale.On the coarsest level, see e.g. ), and on introducSince the convolution with the gradient of a function is equal to the gradient of the convolution, the filtering can be efficiently performed by first convolving with a Gaussian to smooth the image and then computing the gradient. The extraction of ridges of maxima is performed by looking for local maxima in the gradient direction. Additionally, the edge pixels are thresholded using two thresholds in order to reduce 'streaking', that is the subdivision of edges into short segments, while simultaneously reducing the probability to extract isolated edge points.An implementation of the Canny edge detector thus amounts to the following steps 21. Smooth the image by convolving with a Gaussian of variance σ.2. Compute the gradient of the smoothed image, and compute its magnitude and direction.3. Non-maximal suppression: Select the pixels where the gradient magnitude has a local maximum in the direction of the gradient.4. Using two specified thresholds, T1 and T2, with T1 <T2, mark selected pixels with gradient magnitude larger than T2 as 'strong', and pixels with magnitude between T1 and T2 as 'weak'.5. Select all strong pixels, and all weak pixels that are connected to strong pixels horizontally, vertically or diagonally.It is interesting to note that  emphasizω, the width σ and the direction angle θ, is given byA 2D odd Gabor filter, characterized by the angular frequency θ) for two different choices of σ and ω is shown in figure ω, σ and θ may be chosen freely for Gabor filters, while they are fixed to discrete values for the curvelets, with the frequency and width given by the curvelet level and the choice of directions given by the partition of the frequency space. Thus, the Gabor filters are more flexible, which is both an advantage and a disadvantage. The advantage is obviously that the parameters may be chosen to optimize performance for different applications, while the disadvantage is that the generality makes it difficult to find a set of parameters that actually works, making the filters more complicated to use.The 1D profile of this filter  so that σ·ω = 1 is optimal for detection of step edges by convolution with odd Gabor filters. With this choice, the Gabor filter is very similar to the derivative of a Gaussian used in the Canny algorithm, and it is therefore not surprising that this choice gives good results for step edges. To select the angle θ, they suggested estimating the edge direction at each point of an image from the gradient of the smoothed image and then evaluate the convolution with a Gabor filter using the gradient direction for θ. The edges were then marked as pixels of local maxima in filter response. This method is used in the examples in the Results section. argue thσ·ω ≈ 1.25, to detect grain boundaries in electron microscopy images of metals and alloys. They were able to detect edges with a wide range of characteristics, both step and ridge edges, and of different widths, but the sizes of the filters had to be tuned manually to the images, and as mentioned above the even Gabor filter is sensitive to background intensity differences. used a cIn this paper we combine the curvelet transform with a Canny edge detector algorithm leading to an edge detection scheme consisting of the following steps:Algorithm 2 (Edge detection using curvelets)1. Apply the fast discrete curvelet transform to the image2. Using one or several levels of curvelet coefficients, extract the maximum magnitude and direction at each location on the finest of the chosen levels of curvelets. This gives the directional field of the edges in the image.3. Apply steps 3 to 5 of the Canny edge detector , with the directional field as input, to extract the edges.4. Extend the extracted edges along the directional field computed in step 2, to connect neighboring edge segments.5. Map the edges onto the original image and perform post-processing suitable to the application. for the fast discrete curvelet transform  to the image in figure A varied from 0 to 100 . The noisy images were then analyzed by our curvelet-based edge detector and by the Canny edge detector, using the same parameter values as in figure In order to assess the sensitivity of our algorithm to noisy data, we investigate how much the extracted edges change when uniform noise is added to the image. To this end, we added uniform random noise in the interval [-We have described a method for edge detection based on the discrete curvelet transform, and have seen that it can be useful for finding edges and elongated structures in images where the edges may not easily be detected using traditional methods. The main advantages of the curvelet transform are that it is a multi-scale transform, which enables us to extract edge information from detail levels of our choice and disregard the other levels, and that it gives us directional information at each point which can be used to improve the edge detection. Discarding the information on the coarser levels implicitly performs a background substraction and makes the method insensitive to background intensity variations in the image, while discarding the finest level efficiently excludes most of the noise. Excluding however the finest level also excludes information about edges with the width of a single or very few pixels. Our method should therefore primarily be applied to images where the edges to be detected show variations on a wider scale than only a few pixels.This is also seen in the precision with which we detect the edges. Since the curvelets are defined on a coarser grid than the original image, the edges we detect will not have pixel precision, but rather the precision of the grid defined by finest selected curvelet level. This makes sense when the edges we search for are several pixels wide.Since a single edge often leaves a trace on several curvelet levels, and since we may be interested in edges of varying width, it makes sense to include several levels of curvelets in the analysis, and indeed the results often improve when including several levels, as we did in the tube formation example.When extracting the directional field, we pick the direction with the maximal curvelet coefficient as in (2), using its magnitude as the magnitude of the directional field, since this gives us a measure of the strength of the edge in that particular direction without influence from edges in other directions. It is often a good idea to perform a running average over three or more neighboring directions before picking the maximum, since an edge may have a direction which is not perfectly aligned with a single predefined curvelet direction, in which case it influences the curvelet coefficients for several directions and these should be added to include the entire effect of the edge. By computing the sum over all directions as in (3), we can also detect areas where there are many weaker edges meeting, and by distinguishing these areas from those found by picking the maximum, we can distinguish between isotropic and highly anisotropic areas of the image, as in figure The plots in figure The comparisons to the Canny edge detector and the Gabor filter-based detector in figures TG developed and programmed the edge detection method and drafted the manuscript. PK conceived the study, contributed to the development of the edge detection method, edited and finalized the manuscript. All authors read and approved the final manuscript.CurveletUtils. Matlab source code for a GUI implementing the edge detection method.Click here for file


The durations and trajectories of our saccadic eye movements are remarkably stereotyped. We have no voluntary control over these properties but they are determined by the movement amplitude and, to a smaller extent, also by the movement direction and initial eye orientation. Here we show that the stereotyped durations and trajectories are optimal for minimizing the variability in saccade endpoints that is caused by motor noise. The optimal duration can be understood from the nature of the motor noise, which is a combination of signal-dependent noise favoring long durations, and constant noise, which prefers short durations. The different durations of horizontal vs. vertical and of centripetal vs. centrifugal saccades, and the somewhat surprising properties of saccades in oblique directions are also accurately predicted by the principle of minimizing movement variability. The simple and sensible principle of minimizing the consequences of motor noise thus explains the full stereotypy of saccadic eye movements. This suggests that saccades are so stereotyped because that is the best strategy to minimize movement errors for an open-loop motor system. We have no voluntary control over the duration and velocity of our saccadic eye movements. Normal saccades are therefore stereotyped and follow the so-called ‘main sequence’ On top of the stereotypy, saccades display a certain level of variability The noise in individual motoneurons is in a good approximation signal-dependent noise (SDN) The aim of this study was to determine how actual saccades relate to the theoretical movements that minimize the consequences of the actual motor noise. The results show that they are very similar, which suggests that saccades are planned in such a way that the movement variability is minimized.We calculated see  the trajWe calculated the optimal duration for horizontal saccades of amplitudes between 0.25 and 30 deg that moved the eye symmetrically about the primary position, the eye's equilibrium position when it looks straight ahead. R of the levels of CN and SDN, and by the mechanical properties of the oculomotor system . Monkey saccades are indeed about a factor two faster than human saccades The mechanical properties of the oculomotor system are less likely to vary much between humans but they differ strongly across different species The duration of a human saccade is not determined by its amplitude only, but it depends also on the movement direction and the initial eye orientation. Centrifugal saccades, which move the eye from the primary position to an eccentric position, have longer durations, lower peak velocities and more asymmetric velocity profiles than centripetal saccades, which return the eye from an eccentric position to the primary position Saccades in the vertical direction have a longer duration than horizontal saccades of the same amplitude What is the duration of saccades in oblique directions? Let us assume that a 10 deg saccade to the right takes 56 ms and an upward one 60 ms. How long does an oblique saccade 10 deg to the right and 10 deg up then take? Since the oblique saccade requires similar horizontal muscle activations as the rightward saccade and comparable vertical muscle activations as the upward saccade, one could expect its duration to be in the range of that of the rightward and upward 10 deg saccades, i.e., 56–60 ms. That would mean that oblique saccades would be ‘superfast’ because they would be faster than horizontal and vertical saccades of the same amplitude . Actual oblique saccades are however not superfast, but their duration compares to that of purely horizontal and vertical saccades of the same amplitude Another feature of oblique saccades is that their horizontal and vertical components have approximately the same duration, even when their amplitudes are different The trajectories of actual oblique saccades can display some curvature We make saccadic eye movements to direct our fovea quickly to objects of interest. Saccades thus serve vision. It can therefore be expected that the saccadic system has evolved in such a way that it serves vision optimally. It is however not obvious what saccades should look like to support vision optimally. Saccade duration seems important for two reasons. First, when we detect an object of interest, such as a possible predator, in our visual periphery, it is important to direct our gaze to that object as fast as possible. Second, because vision is highly degraded during a saccade, vision is served best by making the duration of a saccade as short as possible. It could therefore be expected that saccades have evolved to be as fast as possible Another important feature of saccades is their accuracy. Individual saccades can miss the desired destination as a result of motor noise and uncertainty in the sensed target location. Such errors can have devastating consequences. For instance, a saccade error may preclude the timely identification of a predator (or prey). Our survival may therefore depend on saccade accuracy. This demonstrates that minimizing variability in saccade endpoints is behaviorally relevant because the smaller the variability, the smaller the mean error will be. The relevance goes however further than this. Once an error has been made, a secondary, corrective saccade will be generated. The number and the amplitude of the required secondary saccades, and therefore the average time needed to reach the desired destination, will all be close to minimal when the endpoint variance is minimized. Minimizing the endpoint variance thus indirectly also approximately minimizes the total time with impaired vision during saccades and the time needed to reach the desired gaze direction.Our results show that the full stereotypy of saccade trajectories and durations is optimal for minimizing the variability in saccade endpoints. Given the above-mentioned advantages, we propose that the saccade system has purposefully been optimized to minimize this variability. It is possible that this optimization process has taken place during evolution, but another possibility is that the optimization has occurred during the development of each individual. Some support for the latter option comes from the observation that saccade duration varies somewhat across subjects Which trajectories and durations are optimal depends on the precise properties of the motor noise In addition to explaining what the stereotypy looks like, the principle of minimizing the consequences of motor noise also explains why there is a stereotypy in the first place. Without the stereotypy, duration and velocity would be more variable. Many saccades would therefore be suboptimal and produce large errors. The only way to avoid this is to always choose a trajectory that is close to the optimal trajectory, or, in other words, to produce stereotyped saccades. This may seem a very general principle that must apply to the goal-directed movements of other body parts as well. The movements of many other body parts are however less stereotyped. We have, for instance, voluntary control over the duration and velocity of arm movements. Why are arm movements less stereotyped? This could be related to their longer duration, which makes it possible to correct movements online on the basis of sensory feedback Although we have shown that a single, simple principle can explain the full stereotypy of saccade durations and velocity profiles, there is one aspect of saccade trajectories that we did not consider. The torsion of the eye was assumed to obey Listing's law. In principle, however, torsion is not constrained to obey Listing's law but it is also a free parameter. It is unknown why the actual eye orientation does obey Listing's law In summary, we have shown that the stereotyped durations and velocities of saccadic eye movements are optimal for minimizing the variability in saccade endpoints caused by motor noise. This suggests that the saccade system has purposefully been optimized to minimize the consequences of motor noise. This optimization process could have taken place during evolution, but it is more likely that it takes place during the development of each individual. A key element of the study is that we minimized the consequences of the recently estimated actual motor noise, rather than that we, incorrectly, assumed signal-dependent noise. This study therefore stresses that, in studies in which motor noise plays a role, it is very important to make correct assumptions about the properties of this noise.n̂ from the primary position; Ω and I = 2.00×10−7 kg·m2B, and the stiffness K were chosen such that (1) corresponds to an overdamped system with time constants of 224 and 13 ms We used the same three-dimensional model of the oculomotor system that we used to estimate the properties of motor noise τ. This vector is multiplied by matrix MR, that describes a rotation of angle n̂The right hand side of (1) represents the torque generated by the extraocular muscles. These muscles were assumed to form three pairs with orthogonal insertions on the globe. The net torques generated by these muscle pairs form the vector mt = 10 ms τ and the aggregate motor command τ. We next expressed eye orientation as three-dimensional rotation vectors rr ˙, r, respectively, and J is a constant. Equation (2) describes a three-dimensional, linear, overdamped system with time constants t1 = 224 ms, t2 = 13 ms and t3 = 10 ms, and with input The muscles produce torques because they receive motor commands from their motoneurons. We modeled the muscles as first order lowpass filters with a time constant of A and B are:When we define the state vector as: CNk and CNk = 1.37·10−5kgm2s−2 and SDNk = 0.172 n1∶n2∶n3 = 1.41∶1.41∶1.00 to reflect the different muscle strengths We assumed that motor commands have constant noise (CN) and signal-dependent noise (SDN) in their magnitude FWe calculated optimal saccade trajectories that, on average, bring the eye to the target with minimal variance in eye orientation, summed over the horizontal, vertical and torsional components, and averaged over a post-movement fixation interval M of a saccade as the duration of its pulse component. Let Ft be the time in the interval , then the variance in the fixation interval  in component i resulting from SDN is Fu is the fixation command required to keep the eye still at the target position, and p(t) is the impulse response function which for all three components is:i resulting from CN is:F is also irrelevant for the cost function, so the cost can be simplified to:The shape of a saccade trajectory is fully determined by the shape of the motor command. Motor commands of saccades are generally assumed to consist of two parts. The pulse component consists of large signals that set the eye into motion and bring it to its end position. The smaller signals of the step component are then required to keep the eye stationary after the movement. We define the duration x0 at t = 0 to final state xf at t = M. Substituting the initial and final states in (5) defines the nine constraints as:H = f+ΓTg be the augmented cost function, where Γ = T is a vector of nine Lagrange multipliers γi. Then, the solution of the optimization problem can be found by solving the Euler-Lagrange equation M, is:Λ is a nine dimensional vector of constants λi that are determined by the nine constraint equations. This solution is identical to the solution if there were no CN. This means that, for a given M, the optimal trajectory in the presence of both SDN and CN is identical to the optimal trajectory in the presence of SDN only, and that was found previously The optimization problem is further specified by the constraints that the eye moves from initial state i can be shown to be: i = 1, 2, 3 and with: D = I00(I122−I11I22)+I01(I01I22−2I02I12)+I022I11, and: pi)((t) is the ith temporal derivative of p(t).The solution for the λThe trajectories of the optimal movements can be found by substituting (14) into (5).M, we consider M as a free endpoint and apply the transversality condition M that has a single solution. The equation shows that the optimal duration depends on the ratio R = CNk/SDNk of the levels of CN and SDN, not on these levels themselves. The optimal duration depends also on the mechanics of the oculomotor system (via the impulse response function), on the fixation period F (via FQ(M)) and, of course, on the initial and final eye orientation (which determine the λi).To find the optimal duration x0 and xf were zero. As a result, the full optimal trajectories appeared to obey Listing's law. We also assumed that the velocity and acceleration at movement onset and offset were zero .We assumed that the eye orientations at movement onset and offset obeyed Listing's law. This means that the first elements of ijI cannot be solved analytically. These integrals were solved numerically using a time-step of 0.025 ms. The optimal duration was determined by solving (16) numerically, also using a time-step of 0.025 ms. The post-movement fixation interval F was set to 50 ms. Making this interval longer led to negligible changes to the optimal durations and trajectories.Although we derived expressions for the optimal trajectories, these trajectories could not be calculated in closed form because the integrals We assumed that motor noise is a combination of SDN and CN. Actual saccades, however, also display temporal variations, which are simultaneous variations in duration and velocity across repeated movements, such that the saccade amplitude is constant


There have been great advances in the conservative and surgical treatment for adolescent idiopathic scoliosis in the last few decades. The challenge for the physician is the decision for the optimal time to institute therapy for the individual child. This makes an understanding of the natural history and risk factors for curve progression of significant importance. Reported rates of curve progression vary from 1.6% for skeletally mature children with a small curve magnitude to 68% for skeletally immature children with larger curve magnitudes. Although the patient's age at presentation, the Risser sign, the patient's menarchal status and the magnitude of the curve have been described as risk factors for curve progression, there is evidence that the absolute curve magnitude at presentation may be most predictive of progression in the long term. A curve magnitude of 25° at presentation may be predictive of a greater risk of curve progression. Advances in research may unlock novel predictive factors, which are based on the underlying pathogenesis of this disorder. Adolescent idiopathic scoliosis (AIS) is defined by the Scoliosis Research Society as scoliosis whose onset occurs after ten years of age and whose cause is essentially unknown. It is a relatively common condition among adolescents.et al., in a school screening study, reported prevalence rates of 0.4 and 2.2% in English girls of 9 to 11 years of age and 12 to 14 years of age, respectively. The boys had prevalence rates of 0.1 and 0.3% in the two age groups.4AIS has been described as having a prevalence of about 0.35 to 13%, depending on the defined Cobb's angles, screening age and sex.et al. reported on the results of screening of 82,900 Greek schoolchildren of 9 to 14 years old during a 1-year prospective study. He reported a prevalence of 1.7% in his study. In total, 2.6% of the girls and 0.9% of the boys had radiographic evidence of a Cobb's angle of 10° or more. He also described the prevalence to vary according to age. A total of 0.07% of the children had scoliosis by the age of 9, 0.2% by age 10 and 0.4% at age 14.5Soucacos et al., reporting on the prevalence of idiopathic scoliosis in Singaporean school children, found that 0.04% of the children had scoliosis by the age of 7, 0.19% by the age of 10 and 1.44% by the age of 14, while the overall predicted prevalence rate for children 9 to 14 years of age  was 0.78% (1.23% in girls and 0.33% in boys).6Wong et al.et al.If left undetected and untreated, AIS can lead to many potential complications. Nachemson,Hence, the natural history of AIS in the long-term would be better analyzed in studies that only include patients who have AIS. Weinstein and Ponseti reported on the long-term outcome of a group of 194 patients with untreated AIS. The patients were an average of 53 years of age and out of the 194, all but 4 were normally active. About 21% had mild psychological reactions to their deformity, such as unwillingness to wear tight fitting clothing or a bathing suit. Backache was described as being somewhat more common compared to a matched group of 100 patients but was not disabling. They did not find an increased mortality rate compared to a match group and cor pulmonale was implicated as a cause of death only in one patient.11There have been significant improvements in both the conservative as well as operative treatment of AIS in the last few decades. However, it has become apparent that despite these advances, the success of treatment still depends on early detection of this condition. Both conservative treatments as well as spinal instrumentation have the best results before the curve has progressed to a large magnitude.This has been an important reason behind the institution of screening programs in many countries that allow for the early detection of curves of smaller magnitude.3The decision on the optimal time to institute bracing or surgical treatment as well as when observation of the curve alone is sufficient can only be confidently made with an understanding of the natural history of adolescent idiopathic scoliosis. However, accurate prediction of curve progression is still not available. Reported rates of curve progression have varied considerably . There iThis is compounded by the issue of differences in the criteria for progression, the institution of treatment during follow-up and the length of follow-up. As a result, it is not clear to what extent these factors may be used to accurately predict the natural history of AIS. To date, there has been no agreement and definitive guidelines have yet to be established in the prediction of curve progression and when to institute treatment for each child.et al., who studied 474 children with AIS defined at a Cobb's angle of 5° or more, reported that spontaneous improvement was observed in approximately 22% of those on follow-up for an average of one year. In the same study, he reported only a 5% incidence of progression of an average of 7°. There was also no difference in the age, gender or curve location between those that improved or progressed.12Indeed, it has been reported that a significant number of idiopathic curves may actually improve during the follow-up period. Brooks et al. reported a 14.7% incidence of curve progression defined as an increase in Cobb's angle of 5° or more. The authors also reported that an unexpected 9.5% of children in his study showed complete spontaneous resolution of the scoliotic curve, while over 35% of the patients with a left thoracic or left and right thoracolumbar curves showed a spontaneous decrease in the magnitude of their curve of at least 10°. In this study, the authors found that the factors that were most associated with the natural history of the scoliotic curve were gender, curve pattern and maturity. They found that girls showed a higher incidence of progression overall and that the difference was even more pronounced in curves that progressed between 5 and 10°. In addition, curves that developed before menarche were found to have almost twice as great a risk for progression.5Soucacos et al. followed 603 children for at least 2 years and reported a 6.8% progression of 5° or more. In addition, 15.4% of the skeletally immature girls with initial curves of 10° or more progressed but he also observed that there was no progression in 20% of the skeletally immature curves with an initial magnitude of 20° or more.2In cases detected through school screening, Rougala Lonstein and Carlson retrospectively reviewed the progression of 727 children with idiopathic scoliosis defined by a single radiograph showing scoliosis of 29° or less. Due to the varying definitions of curve progression in previous studies which varied from 5 to 10°, Lonstein and Carlson instead defined curve progression in their study as an initial curve of 19° or less that increased at least 10° with the final magnitude being greater than 20° and an initial curve of between 25 and 29° that increased by 5° or more. They reported a curve progression rate of 23.2% and found that curve magnitude, skeletal immaturity and curve pattern were associated with progression.13The various factors associated with progression have also been combined for more useful model for prediction of curve progression. Lonstein and Carlson used a combination of curve magnitude and the Risser sign to calculate the probability of curve progression. They found that for children with a Risser grade of 0–1, those with a curve magnitude of 5–19° had a 22% progression rate compared to a 68% progression rate for those with a curve magnitude of 20–29°. Similarly, for children with a Risser grade of 2–4, those with a curve magnitude of 20–29° had a progression rate of 23%, while those with a curve magnitude of 5–19° had only a 1.6% progression rateet al. performed genetic association studies to investigate variation of the melatonin receptor 1A (MTNR1A) and 1B genes in girls with adolescent idiopathic scoliosis (AIS) patients compared to normal controls. He found an association with the 1B gene but not the 1A gene polymorphism.16et al. analyzed the estrogen receptor gene in 304 girls with idiopathic scoliosis whom he followed up till skeletal maturity. He found that the Xbal site polymorphism was associated with curve progression defined as an increase of more than 5° from initial evaluation.18There is an increasing evidence that genetic factors have a part to play in the development in idiopathic scoliosis.More recently, there has been progress in using a combination of multiple genes associated with the development of severe curves to predict curve progression. Ogilvie has developed a saliva-based test for a combination of multiple genetic markers that have been linked with curve progression in clinical trials as a prognostic test for AIS. This allows for the calculation of a quantitative score that correlates with a low, medium or high risk of curve progression.19Once the child has attained skeletal maturity, it was generally thought that the curves are less likely to progress. However, this may not always be the case. It is now established that curves due to idiopathic scoliosis do not necessarily stop progressing after skeletal maturity. In a long-term follow-up study of patients with idiopathic scoliosis, Collis and Ponseti found that curves of a larger degree did increase after skeletal maturity.21Given the varying definitions of curve progression, this suggests that curves with a Cobb's angle of 30° are an important threshold magnitude and may serve as an endpoint for prediction of curve progression rather than predefined units of curve progression quoted in previous studies. In addition, it must be appreciated that the various associated factors and predictions described only apply to the likelihood of a curve progressing in adolescence. They are only averages and correlations and do not allow us to answer the key issue of how much the curve of an individual child is going to progress.In a recent study, Tan and Wong reported on a group of 279 patients with idiopathic scoliosis detected by school screening, and who were followed-up until skeletal maturity using a 30° Cobb's angle at skeletal maturity as a threshold instead of predefined units of curve progression during shorter periods of growth. They found that an initial Cobb's angle of 25° was the most predictive factor for curve progression to this threshold magnitude. Initial age, gender and pubertal status were less important prognostic factors.Adolescent idiopathic scoliosis is a relatively common condition that if left untreated can lead to significant morbidity and possibly mortality. Early detection of curves has been facilitated by school-based screening but has resulted in a need for the understanding of the natural history and reliable prediction of curve progression to decide on the appropriate treatment and timing of intervention. Individual factors such as the patient's age at presentation, the Risser sign, the patient's menarchal status and the magnitude of the curve have been used as predictive factors as well as using a combination of factors to predict curve progression. However, the relative importance of each factor and how they may interact is as yet not defined. Based on current evidence, we recommend closer follow-up of skeletally immature children with a curve magnitude of 25° or more at presentation and also continuing to follow-up children with a curve magnitude of 30° or more even after skeletal maturity. Moving forward, the ideal prognostication model needs to be one that allows for prediction of the amount of progression that is likely to be observed over the whole period of remaining growth and should be individualized to the child. This may only be attained by the introduction of novel predictive factors rather than factors that are surrogate markers of potential remaining skeletal growth that have been used in the literature thus far. This may only be realized by unlocking the underlying pathogenesis of AIS.