Development of effective and plausible numerical tools is an imperative task for thorough studies of nonlinear dynamics in life science applications.We have developed a complementary suite of computational tools for two-parameter screening of dynamics in neuronal models. We test a ‘brute-force’ effectiveness of neuroscience plausible techniques specifically tailored for the examination of temporal characteristics, such duty cycle of bursting, interspike interval, spike number deviation in the phenomenological Hindmarsh-Rose model of a bursting neuron and compare the results obtained by calculus-based tools for evaluations of an entire spectrum of Lyapunov exponents broadly employed in studies of nonlinear systems.We have found that the results obtained either way agree exceptionally well, and can identify and differentiate between various fine structures of complex dynamics and underlying global bifurcations in this exemplary model. Our future planes are to enhance the applicability of this computational suite for understanding of polyrhythmic bursting patterns and their functional transformations in small networks. Individual and networked neurons can generate various complex oscillations known as bursting, formed by alternating fast repetitive spiking and quiescent or subthreshold oscillatory phases. Bursting is a manifestation of composite, multiple time scale dynamics observed in various fields of science as diverse as food chain ecosystems, nonlinear optics, medical studies of the human immune system, and neuroscience. The role of bursting is especially important for rhythmic movements determined by Central Pattern Generators (CPG). Many CPGs can be multifunctional and produce polyrhythmic bursting patterns on distinct time scales, like fast swimming and slow crawling in leeches . Such CPIn mathematical neuroscience a deterministic description of endogenously oscillatory activities, like two-time scale bursting, is done by revealing generic properties of mathematical and realistic models of neurons; the latter are derived through the Hodgkin-Huxley formalism for gating variables. Either bursting model falls into a class of dynamical systems with at least two time scales, known as slow-fast systems. Configurations and classification schemes for bursting activities in neuronal models first proposed in  and extefold/homoclinic and fold/Hopf, as this would indicate that the terminal phases of the fast spiking and slow quiescent periods are determined, respectively, by a homoclinic bifurcation of a saddle equilibrium state, or a supercritical Andronov-Hopf bifurcation, along with a saddle-node bifurcation of equilibria, respectively, which all occur in the fast subsystem of the model. In the next section we will examine the transition bifurcation patterns between these types of bursting. The types of the static bursting configurations in the Hindmarsh-Rose model shown in Figures These manifolds, especially their stable branches, can be easily traced and visualized in the phase space by utilizing the slow variable as a sweeping parameter in the decoupled fast subsystem. Far from bifurcations, this slow-fast dissection approach allows for exhaustive simplifications that let one treat the dynamics of the full model as on overlay the uncorrelated dynamics of its fast subsystem mediated by repetitive passages of the slow variable.as long as it stays away from a bifurcation that is due to reciprocal interactions of the dynamics of both subsystems. Such a bifurcation, underlying a bursting transition(s), gives rise to the emergence of dynamical phenomena that can only occur in the full system. For example, this occurs when the dynamics of the fast subsystem falls to the time scale of the slow subsystem, particularly near saddle-node and homoclinic bifurcations. A classic example is the onset of chaotic dynamics of finite subshift shown by D. Terman [ The slow-fast dissection has been proven to work very well for a low-order model of a bursting neuron . Terman  at the t. Terman -15. In a. Terman  gives an. Terman  prolongi. Terman ,18 playitranscends the existing static classification schemes based solely on slow-fast dissection. Complex dynamics, including quick period doubling cascades in square wave bursters ,20 can aI in (1), a neuron model should demonstrate, migrate, and switch flexibly between various types of activities such as quiescence, tonic spiking and bursting. In addition, nonlinearity of the model can often imply bi- or multi-stability of several co-existing activities at the same parameter values. Bistability of coexisting oscillatory patterns originate near global bifurcations taking place in the model. Multistability is well noticeable when a targeted activity can be robustly selected by choosing other initial conditions or by temporal perturbations, like applied external current. Having ascertained such a global picture we can make consistent predictions for determining basic principles of the functioning of coupled neurons on networks where they receive mixed, inhibitory and excitatory perturbations from other neurons and synergetically reciprocate.In-depth understanding of the generic mechanisms combined into a broad global picture on the transition patterns between the activity types in typical models of individual neurons presents a fundamental challenge for the theory of applied dynamical systems. In response to variations of intrinsic parameters, or an external applied current, like limit solutions of the system. Nevertheless, if performed extensively the brute force approach reveals adequately the backbone of the bifurcation structure of the model, which can be further enhanced and complemented with detailed bifurcation analysis that would provide the finishing touches, in the form of bifurcations curves, to the initial brute force diagram. We point out that transformations reshaping bursting waveforms en route toward the tonic spiking activity are caused by atypical bifurcations due to the presence of two, or more time scales in bursting. Because of that bifurcations of stiff bursting solutions, especially irregular ones, are hard to trace down by parameter continuation software packages such as CONTENT and AUTO-based packages, which are specifically designed mainly for explorations of equilibria and ‘typical’ periodic orbits.The applied dynamical systems community has developed a universal and versatile suite of computational tools and techniques for comprehensive examination of diverse nonlinear systems of various origins including life sciences. These tools allow a researcher to mono- and bi-parametrically scan the model in question to search for specific transformations corresponding to local and global bifurcations that often hard to detect by standard means. One approach to an initial examination of an unknown model, often called ‘brute-force’ approach, is to evaluate the largest Lyapunov exponent. This approach has long been widely utilized in nonlinear dynamics for initial detection of bifurcations of steady state and oscillatory solutions. The brute-force approach contrasts drastically from examination of fine bifurcation structure of  The primary goal of this paper is to demonstrate that straightforward methods used in neuroscience experimental studies can be as effective as conventional tools, based on the bifurcation and Lyapunov exponent theory, employed in nonlinear dynamics studies. In this paper, we revisit and examine transformations of various oscillatory activity types in the phenomenological Hindmarsh-Rose model of bursting neurons, viewed, so to speak, through the prism of neuroscience plausible methods. Next we compare our findings with the results obtained using the evaluation of a maximal Lyapunov exponent that was presented in detail in ,28 which The phenomenological system of ODEs proposed by Hindmarsh and Rose ,30 for mx is treated as the membrane potential, while y and z describe some fast and slow gating variables for ionic currents, respectively. Slow ‘activation’ of z is due to the small parameter b, and ‘external,’ I, bifurcation parameters the dynamics of the model are sensitive to variations to other parameters: ε being treated as a rate of activation for some current, and I in (1), a neuron model should demonstrate, migrate, and switch flexibly between various types of activities such as quiescence, tonic spiking and bursting. here, In this section we will brief the core of the numerical techniques employed in the analysis of the HR model. We will start with the specifics of the numerical integration of the differential equations of the model (1).http://gme.unizar.es/software/tides[nth degree Taylor series of f is to be a smooth or analytical function). So, denoting  There are a plethora of high quality numerical integrators that have been created by numerical ODEs specialists. This study utilizes a recently developed freeware library TIDES  available at are/tides. TIDES i Therefore, the problem is reduced to the determination of the Taylor coefficients tails in ).DOPRI853 developed by Hairer and Wanner [ The Taylor method has several unique features ,33. One accurate  than thed Wanner . Note thd Wanner . As rema As mentioned above, the continuous output generated by the integrator based on the Taylor series method is able to detect accurately and effectively various instantaneous events such as whether the phase point hits a cross-section or reaches some critical value like voltage maximum/minimum, or the number of spikes per burst approaches the sought value, which is the underlying idea for the spike-counting (SC) technique . In comb The other technique that we use in this paper is the computation of the Lyapunov exponents for solutions of the HR model. In bounded systems the existence of a positive Lyapunov exponent is associated with chaotic motions, and any orbit on a compact set that does not converge to an equilibrium point has at least one zero Lyapunov exponent. Accordingly, computation of the largest Lyapunov exponents yields important information about the kind of orbits present in the system. The corresponding algorithm for the computation of the Lyapunov spectrum is a combination of the classical methods proposed by ,37 and tb and I; both are in charge of transformations for the intrinsic structure of the fast subsystem in the HR model. Next we perform the two-parameter sweep or screen of the model to collect vital data representing the time and parameter transformations of the single ‘voltage’ x-variable of the neuronal model. Further data mining will be carried out to extract quantitative and qualitative information about the dynamical variability of the model, bifurcations of its solutions, and so on. In Figures 6 simulations. The HR model can exhibit a plethora of dynamical activities at different parameter values. Consequently, obtaining a comprehensive understanding of the multi-parametric evolution of a system like the HR model is no easy task, and hence most of the parameters are fixed in the model. In this section, the free bifurcation parameters to be varied are  Figure orted in . The colb and I change the structure of the fast subsystem of the HR model so that the homoclinic bifurcation is no longer transverse in the z-parametric cutaway in the singular limit [ Another interesting behavioral phenomenon reshaping the type of bursting occurs in the top-left corner of the diagrams. In this region of bursting with a high number of spikes transforms into bursting with a drastically lower number of spikes per burst. To elucidate what happens on the border between these regions, we visually examined the orbits of the model. We found the border corresponding to the transition between square-wave and plateau-like bursting . This bursting orbit has five large spikes followed by a single short spike. After this short spike disappears the bursting orbit steadily exhibits five spikes. The process occurs at every spike-deletion bifurcation, or spike adding bifurcation, if instead the parameter b is decreased. We have pointed out the changes in the color representation  of the duty-cycle in Figure  Indeed, the fact that the interspike interval grows by the end of the burst is a signature of square-wave bursters. Recall that these bursters are also code-named fold/homoclinic meaning that the spiking, slow-motion manifold is terminated through a homoclinic bifurcation of the saddle that occurs in the fast subsystem of the model, see Figure l models ,24,40,42l models ,18.ε. In the neuroscience context, ε can be treated as the reciprocal of τ, which determines the (in)activation rate of the slow current in a neuronal model. For the sake of consistency we will first screen the model in the x-direction . As x-maxima of the short waveform. As was pointed out earlier such a coexistence of ε, the phase point can be dragged along the saddle branch up to the upper fold on ε is increased. We would like to point out that multistability is a by-product of the nonlinearity in the system. Multistability has been reported in several neuronal systems both experimentally and computationally, including individual neurons and their models, as well as in neural networks and multifunctional central pattern generators [ Figure on model . Recall nerators . MultistI. This brings in one remaining question we would like to addresses in the paper: will this property of self-similarity of the band structure persist for smaller values of ε? Our findings are summarized in Figure ε. We can deduce from the diagrams that the most interesting, in terms of dynamics, region is contained between two curves (indicated by white dots). Within this region there are several diagonal bands corresponding to bursting orbits with different numbers of spikes. This plot explains the fact that a small ε-parametric cut will nevertheless reveal the band structure, observed in Figures I or Another peculiar observation related to the band structure in the  As of today, the Hindmarsh-Rose remains justifiably one of the most popular mathematical models, that describes, qualitatively well, the dynamics of a certain class of neuronal models derived using the Hodgkin-Huxley formalism. The model has been carefully analyzed using various mathematical and computational tools, and has been viewed though the prisms of the advanced bifurcation theory, geometric methods of slow-fast dynamical systems to reveal multiple peculiar qualitative features. Various brute force approaches ,28 have x, treated as a voltage across the cell membrane, or more specifically, the corresponding voltage waveforms. The list includes the spike-counting approach, the evaluations of inter-spike interval, and duty cycles of bursting, which is a ratio of the burst duration (active phase) over the total burst period. We confirmed our findings based on these methods with ‘calculus’-based simulations for the whole spectrum of the Lyapunov exponents calculated for solutions of a system of ordinary differential equations like the Hindmarsh-Rose model. Our verdict is that both approaches consistently demonstrate very good agreements. The methods allow us to give detailed explanations for various global bifurcations in the model, including various spike-addition/deletion cascades, phenomena of bistability, and various transitions between types of bursting: square-wave and plateau-like. This ensures that this toolkit of combined neuroscience-native methods and calculi-based algorithms will yield effective and timely insights for pilot studies of new, previously unidentified, in sense of dynamics and bifurcations, models of individual neurons, as well as other cells, such as myocytes - cardiac tissue cell. We showed in [ In this paper we have tested other computationally effective tools tailored specifically for models originating in neuroscience, including 1D and 2D parametric screenings of the complex dynamics of the model aimed to bridge notions common for neuroscience with the accurate mathematics-based findings. Our computational toolkit includes several methods for examining of temporal characteristics of a single variable, howed in  that the It is in our future plans to broaden the applicability of these proposed computational tools for studies of neuronal networks, especially for multifunctional central pattern generators comprised of several neurons ,46. SuchThe authors declare that they have no competing interests.


We propose to combine a biomimetic navigation model which solves a simultaneous localization and mapping task with a biomimetic sonar mounted on a mobile robot to address two related questions. First, can robotic sonar sensing lead to intelligent interactions with complex environments? Second, can we model sonar based spatial orientation and the construction of spatial maps by bats? To address these questions we adapt the mapping module of RatSLAM, a previously published navigation system based on computational models of the rodent hippocampus. We analyze the performance of the proposed robotic implementation operating in the real world. We conclude that the biomimetic navigation model operating on the information from the biomimetic sonar allows an autonomous agent to map unmodified (office) environments efficiently and consistently. Furthermore, these results also show that successful navigation does not require the readings of the biomimetic sonar to be interpreted in terms of individual objects/landmarks in the environment. We argue that the system has applications in robotics as well as in the field of biology as a simple, first order, model for sonar based spatial orientation and map building. Bats have evolved sophisticated echolocation systems The problem of building spatial maps autonomously has been documented extensively in the robotics literature. Algorithms for Simultaneous Localization and Mapping (SLAM) combine information from proprioceptive sensors  and exteroceptive sensors  to estimate the position of the robot. At the same time a feature-based map of the environment is constructed, i.e. mapping objects onto parametric representations such as points, lines, circles, corners, etc We argue Parallel to the traditional robotic simultaneous localization and mapping systems based on probabilistic methods, biologically inspired solutions have also been proposed. A highly successful example thereof is the RatSLAM system The BatSLAM model for map building based on biomimetic sonar should be viewed as an experiment in synthetic psychology For the exteroceptive sensory input to BatSLAM we propose to capture the echo signals through microphones inserted in replica's of real bat pinnae. See the These results show that informative spectrospatial cues are introduced both at emission and reception. Furthermore, comparing the monaural ERTF patterns with the interaural intensity difference patterns indicates that a binaural system can extract spatial information from a larger field of view. Finally, the ERTF results shown in Through the ERTF the sonar system maps the set of echoes from three dimensional reflector positions onto a two dimensional time-frequency representation, i.e. the cochleogram see . The smoIn the first experiment, we quantify the local accuracy of the metric information in the experience map by mapping a controlled small-scale environment. In addition to the biomimetic sonar, a Hagisonic StarGazer To evaluate the large-scale mapping performance of BatSLAM we performed a second mapping experiment in a typical, unmodified, office environment. The biomimetic sonar system mounted on a mobile robot  was driven through the environment for approximately 45 minutes, while collecting sonar measurements. Consecutive measurements are separated by a mean displacement of 13.5 cm. The large scale of the environment precluded the use of the indoor localization system in this experiment. During the large-scale mapping experiment, the system collected 6000 sonar snapshots. Based on these 6000 sonar measurements, the BatSLAM system generated 4300 experiences in the experience map and stored 3300 local view templates. To allow a ‘fingerprint’ based mapping scheme like BatSLAM to construct accurate and stable maps these local view templates have to fulfill two conditions.First, the local view templates should be distinctive of particular places in the environment. As shown by the re-use histogram in In addition, the local view templates should have properties that change continuously in the neighborhood of the place they are distinctive of. This second condition is important for efficient loop-closure detection. It guarantees that similar robot poses, i.e. robot position and orientation, give rise to similar cochleograms thereby facilitating the recognition of already existing experiences, i.e. loop-closure. Large ambiguous regions could be a problem for BatSLAM as the robot can only rely on odometric information while moving within such an ambiguous region. However, from the results shown in From the results of the mapping experiments we conclude that BatSLAM is able to create consistent maps of large scale, unmodified, office environments. Furthermore, the constructed maps converge over time to fairly accurate metric maps supporting efficient, i.e. distance based, navigation through such environments. These results mirror the conclusions reached in We do not make any claims about the biological accuracy of the details of the proposed implementation for bat environment mapping. However, we propose that BatSLAM, being both explicit and biologically plausible, is a useful first-order model of bat localization and mapping. Clearly, the information required, i.e. the steering commands and the output of the cochlear model A necessary condition for an agent to follow a route or localize itself on a map of an environment is that the agent has access to unique descriptions of the places in that environment. In Extrapolating from the uniqueness of the complex cochleogram shown in Many simplifications are involved in BatSLAM: 2D paths instead of 3D paths through the environment, an office environment instead of an outdoor environment, functional models of hippocampal structures and of the bat's sonar system instead of detailed copies. Nevertheless, we would like to argue that its success makes this model both a robust and easy to implement SLAM system to be used in real-life robotic applications as well as a useful, first order, computational model of the map building processes supporting middle-scale navigation in bats.This section provides details on the computational model implemented in the BatSLAM system. We start with an overview of the biomimetic sonar system and the signal processing applied to the incoming echo signals. For a detailed description and analysis of its performance the reader is referred to Micronycteris microtis scaled by a factor of 1.5 with respect to their true dimensions. Each microphone signal is amplified using a 6th order Butterworth anti-alias active filter with cutoff frequency at 150 kHz. Finally, the right and left microphone signals are digitized using 12bit analog to digital converters running at 500 kS/sec. The pick-up of the sonar emission by the receivers is removed by selecting the part of the microphone signal after the emission has stopped.The different functional blocks that make up the custom-made sonar system are depicted in Micronycteris microtisMicronycteris microtis outer ears were obtained by micro-CT imaging For the emitted signal we chose a hyperbolic chirp, i.e. a sinusoidal signal of 3 ms duration whose instantaneous frequency is hyperbolically swept from an upper frequency of 100 kHz down to a lower frequency of 20 kHz Click here for additional data file.


We present a new method for removing artifacts in electroencephalography (EEG) records during Galvanic Vestibular Stimulation (GVS). The main challenge in exploiting GVS is to understand how the stimulus acts as an input to brain. We used EEG to monitor the brain and elicit the GVS reflexes. However, GVS current distribution throughout the scalp generates an artifact on EEG signals. We need to eliminate this artifact to be able to analyze the EEG signals during GVS. We propose a novel method to estimate the contribution of the GVS current in the EEG signals at each electrode by combining time-series regression methods with wavelet decomposition methods. We use wavelet transform to project the recorded EEG signal into various frequency bands and then estimate the GVS current distribution in each frequency band. The proposed method was optimized using simulated signals, and its performance was compared to well-accepted artifact removal methods such as ICA-based methods and adaptive filters. The results show that the proposed method has better performance in removing GVS artifacts, compared to the others. Using the proposed method, a higher signal to artifact ratio of −1.625 dB was achieved, which outperformed other methods such as ICA-based methods, regression methods, and adaptive filters. Brain stimulation by means of electrical currents has been employed in neurological studies for therapy purposes for many years –5. Howev Considering that the frequency spectra of the neural signals and GVS artifacts overlap, filtering the frequency components of GVS artifacts results in the loss of the original neural signals. The four major EEG frequency bands are Delta (the lowest frequency band up to 4 Hz), Theta (4 Hz to 8 Hz), Alpha (8 Hz to 12 Hz), and Beta (12 Hz to 30 Hz). In order to analyze and understand the effect of GVS on EEG patterns, it is essential to be able to remove the artifact signals from the frequency band of interest, before establishing any GVS-EEG interaction models.There are various methods to remove different types of artifacts, such as myogenic artifacts –9, oculaWe propose a novel method for GVS artifacts removal by combining time-series regression methods and wavelet decomposition methods. To enhance the precision of the artifact estimation using regression models, the models should account for the complex behavior of the GVS interactions in the frequency domain. So we decomposed the recorded EEG and GVS signals into different frequency bands and then used regression models to estimate the GVS artifacts in each frequency band. We used multiresolution wavelet analysis to decompose nonstationary EEG signals in the time-frequency plane. Both the discrete wavelet transform (DWT) and the stationary wavelet transform (SWT) algorithms were employed, and the results were compared. To estimate the GVS current distribution through the scalp using time-series regression methods based on biophysical models, we used and compared the performance of different parametric regression models, such as discrete-time polynomials, nonlinear Hammerstein-Wiener, and state-space models.In this study, we firstly used simulated data to assess and optimize the performance of the proposed method using various regression models and different wavelet algorithms. The resulting optimized method was then applied to real data. We compared the results of the proposed method and other methods, such as ICA, using both simulated and real data. This paper is organized as follows:  The EEG recording was carried out with a NeuroScan SynAmps2 system, with 20 electrodes located according to the international 10–20 EEG system  and withf-type power spectrum within a frequency range of 0.1 to 10 Hz and duration of 72 seconds. We kept the amplitude of the delivered stimuli lower than the feeling threshold, in the range of 100 μA to 800 μA, with the root mean square values between 60 μA and 450 μA. The stimulator is equipped with a data acquisition device to record the delivered stimulus, which allows us to make a continuous record of the delivered stimulation current and voltage. We recorded the EEG signals during the stimulation, 60 seconds before and 60 seconds after the stimulation. The EEG data for these experiments were acquired by our collaborator in the Pacific Parkinson's Research Centre. Nine healthy subjects , between the ages of 21 and 53 yr, with no known history of neurological disease or injury, participated in this study. All subjects were asked to relax, remain still, and concentrate on a focal point on the screen in front of them so that less myogenic and ocular artifacts occur. Also, under resting conditions, there are less variations in the head impedance , therefore minimizing the term E[e2(n)].orAmong the various optimization techniques, we chose the Recursive Least-Squares (RLS) and the Least Mean Squares (LMS) for our application. In the section “Comparison of the performance of different artifact removal methods”, we compared the results of adaptive filters with those of the other methods.Welch's method  algorithm can be applied, and the set of basis functions are defined on a “dyadic” grid in the time-scale plane asj governs the amount of scaling and k2j governs the amount of translation or time shifting. The wavelet transform is the inner product of the basis wavelet functions and the signal in the time domain. In the DWT algorithm, the discrete time-domain signal is decomposed into high frequency or details components and low frequency or approximation components through successive low pass and high pass filters. For multi resolution analysis, the original signal is decomposed into an approximation and details parts. The approximation part is decomposed again by iterating this process; thus one signal can be decomposed into many components. The basic DWT algorithm does not preserve translation invariance. Consequently a translation of wavelet coefficients does not necessarily correspond to the same translation of the original signal. This nonstationary property originates from the downsampling operations in the pyramidal algorithm. The algorithm can be modified by inserting 2j − 1 zeros between filters coefficients of the layer j, instead of down-sampling. This modified version of the DWT algorithm is called stationary wavelet transform (SWT), and it can preserve the translation invariance property. In this study, we applied both DWT and SWT, to decompose the EEG signals using different mother wavelets such as Symlet and Daubechies of different orders. Both the GVS current and the simulated EEG signals were decomposed into 12 levels, and thus we have the frequency bands for approximation and detail components, shown in  As shown above, the main GVS frequency components are in the range of 0.1 to 10 Hz. To achieve the best fit between the estimated GVS contribution and the measured EEG at each EEG channel, we broke down the recorded GVS current and the contaminated EEG data into various frequency bands by means of wavelet analysis and estimated the GVS artifacts in each frequency band. Wavelet transform is able to construct a high resolution time-frequency representation of nonstationary signals like EEG signals. In wavelet transform, the signal is decomposed into a set of basis functions, obtained by dilations and shifts of a unique function Independent Component Analysis (ICA) is a statistical method used to extract independent components from a set of measured signals. This method is a special case of the Blind Source Separation methods, where the K channels of the recorded EEG signals (E(t) = e1(t),…, eK(t)) are assumed to be a linear combination of N(N ≤ K) unknown independent sources (S(t) = s1(t),…, sN(t)):M is the unknown mixing matrix defining weights for each source contributions to the EEG signals recorded at each channel. In ICA, the measured K channel EEG signals are taken into an N dimensional space and projected onto a coordinate frame where the data projections are minimally overlapped and maximally independent of each other. There are various algorithms with different approaches to find the independent components, such as minimizing the mutual information or maximizing the joint entropy among the data projections. The ICA algorithm we used in this study is the extended Infomax algorithm [lgorithm  which islgorithm . It useslgorithm  and wideWe tried two approaches to remove the artifact. The first approach is to zero out the artifact signals from the components that account for the GVS parasitic influence and obtain a new cleaned-up source matrix possible . Eventually, we reconstruct ICA-corrected EEG signals as:In the proposed method, we decomposed the EEG and GVS current signals in 12 frequency bands , and theThe estimated GVS artifact frequency components are subtracted from the contaminated EEG frequency components. The wavelet decomposition enables us to focus on the frequency bands of interest and calculate the estimated GVS artifacts in each frequency band independently; thus the regression method can deal better with some nonlinear behaviors of the skin in the frequency domain. This wavelet-based time-frequency analysis approach enhances the performance of the artifact removal method. The cleaned-up signal is reconstructed from the proper frequency components of the estimated GVS signal components in the frequency range of interest . We calculated the correlation coefficients between the GVS signals and the estimated GVS artifacts reconstructed from different frequency bands, and we observed that the regression results improve when we reconstruct the estimated GVS artifact components from corresponding frequency bands separately.Pearson's correlation for the correlation analysis asu(t) is the recorded GVS current and The result of the correlation analysis is tabulated in  The result shows that the correlation between the GVS signal and the estimated GVS artifact significantly increases by using wavelet decomposition method. We applied the wavelet transform to remove frequency components lower than 0.98 Hz and higher than 31.25 Hz, which are not of the main interest, and the correlation between the GVS signal and estimated GVS artifact was increased up to 0.9899.normalized residual sum of squares which is sometime introduced as the normalized quadratic error defined byEo(t) represents the original artifact-free signal and We employed both SWT and DWT algorithms in the proposed artifact removal method. The difference between SWT and DWT algorithms was briefly explained in the wavelet analysis section. We also used various regression models to estimate the GVS artifact. To assess the performance of the proposed method using different algorithms and models, we applied our method to the simulated data and examined the cleaned-up EEG signals in comparison with the original artifact-free EEG signals. For this assessment, not only did we calculate the correlation between the artifact-removed EEG signals and the original artifact-free EEG signals, but also we measured the fitness of the artifact-removed signals based on the Daubechies  and Symlets . Both SWT and DWT were used with these mother wavelets in the proposed artifact removal method and applied to the simulated data. We tabulated the normalized residual sum of squares and the correlation between the artifact-removed signals and the original artifact-free signals in the frequency range lower than 31.25 Hz  as a simple special case of the general polynomial model, Hammerstein-Wiener with the piecewise-linear function, and Space-State models, which were all introduced in the “Regression-based approach” section. We employed these models with different orders in the proposed artifact removal method and applied the proposed method using each of these models to the simulated data. In order to compare the performance, we used SWT with Daubechies 4 to decompose the contaminated signals, estimated the GVS artifact using different models, and then assessed the performance in terms of the correlation and the normalized residual sum of squares between the original artifact-free signal and the artifact-removed signal reconstructed in the frequency range lower than 31.25 Hz. The results are tabulated in  For nonlinear Hammerstein-Wiener models we used the piecewise-linear function and broke down the EEG signal into a number of intervals. We tried a various number of intervals and observed that, with 4 intervals (or less), we could get the highest correlation and the least residual.The results show that, between all those models, both Output-Error and nonlinear Hammerstein-Wiener have better performance. We employed these regression models to maximize the performance of the proposed method, then we applied the proposed method to the real data.We also used two ICA-based methods for removing the artifact: filtering out the artifact components and applying a threshold on the artifact components amplitude to remove the artifact spikes beyond the threshold.To assess the performances of the ICA methods on the simulated data, we calculated both the correlation and the normalized residual sum of squares between the artifact-removed EEG signals and the original artifact-free EEG signals.We compared the ICA-based methods with the proposed methods using the Output-Error and nonlinear Hammerstein-Wiener models order 2, along with 12-level STW decomposition with DB4 mother wavelet Tables  and 7. We applied different artifact removal methods on real EEG data acquired during application of GVS. We used the data from channel O1  of different subjects in EEG/GVS studies. We applied stimulation signals of different amplitudes in our experiments and observed consistent results from these experiments. By calculating the correlation coefficients between the GVS signals and the estimated GVS artifacts, we compared the performance of these methods. First we compare ICA-based, regression-based, and adaptive filters without using the wavelet analysis. Then we use the proposed method where the wavelet analysis was employed to improve the performance of our artifact removal method.The best algorithms for ICA-based methods, best models for regression-based methods and best filters for adaptive filtering methods were selected. Between different ICA algorithms , the extended Infomax showed better results. Between regression-based methods , OE order 2 showed better performance, and between adaptive filters , RLS filter with the forgetting factor of 0.99997, the filter length of 2, LMS filter with the adaptation gain of 0.5, and the filter length of 3 had better performance. We tabulated  the corrThe results show that, between all the above methods, the regression-based methods are able to estimate the GVS artifacts with higher correlation with the original GVS signals. Thus, we employed the regression-based method along with the wavelet analysis in our proposed method to achieve the best performance in removing GVS artifact. The wavelet decomposition method improves the estimation of the GVS artifacts in both correlation performance and robustness. This is due to the separate transfer function estimations for each frequency band, aspect that makes it less prone to nonlinear skin behavior or to other noise sources. Furthermore, with wavelet decomposition, we can filter out the frequency components that are not of interest. Removing those frequency components can improve the results of the regression analysis as well. The cleaned EEG data is reconstructed from the frequency range of interest .Using a correlation analysis, we show how the wavelet-based time-frequency analysis approach enhances the performance of the artifact removal method. We calculated the correlation coefficients between the GVS signals and the estimated GVS artifacts reconstructed from different frequency bands  was applied, and the EEG/GVS data were collected. We compared the performance of the proposed method and the extended Infomax ICA method. The results show that while the performance of the ICA method deteriorates as the GVS amplitude is increased, the proposed method provides a robust performance  and the nonlinear Hammerstein-Wiener order 2 have similar performances, better than the other models.In the previous section, we compared the performance of different methods and observed that how the combining of wavelet decomposition and regression analysis  can imprUsing the proposed method, we can focus on specific frequency bands and remove the GVS artifact with better performance in each frequency band, separately. Figures The results show that for frequency components L6 to L10, which correspond approximately to 8–16 Hz, 4–8 Hz, 2–4 Hz, 1-2 Hz, and 0.5–1 Hz bands, we can achieve higher performance in rejecting the GVS artifacts separately. One of the reasons of the robustness of the method is building separate equivalent transfer functions for the GVS signals for each frequency band which helps in maintaining the performance of the algorithms for a large range of GVS intensity levels and frequency ranges. To illustrate the importance of the wavelet analysis, we depicted the artifact-removed signals using different frequency components Figures , and 14.signal to artifact ratio (SAR). SAR is defined as the ratio of the signal amplitude to the artifact amplitude in decibels (dB). We can achieve an SAR of −1.625 dB in the frequency range of 1 Hz–16 Hz, while, using the frequency components in the range of 1 Hz–32 Hz , we could reconstruct a major portion of the GVS artifact. As an example, for channel 18, around 87% of the GVS artifact was reconstructed , thus weUsing the wavelet decomposition, we were able to reconstruct up to 96% of the GVS artifact components in some frequency bands, especially in the frequency range of the GVS signals .We showed that the use of the wavelet decomposition can improve the time domain regression approach to estimate the GVS artifacts. By means of the combination of the regression and wavelet analysis in the proposed artifact removal method, we were able to focus on different frequency bands and significantly improve the SAR of the contaminated EEG data in specific frequency bands.The proposed method and the ICA-based methods behave differently in rejecting the GVS artifact. We observed a high correlation between the estimated GVS artifacts and the original GVS signals using the proposed method, but we could not obtain a good correlation using the ICA-based methods.As illustrated earlier, we cannot completely remove the GVS contamination in all frequency ranges . Removing the whole GVS artifacts remains a problem for the future approaches.In this study we also observed that nonlinear Hammerstein-Wiener model of the second order, using piecewise-linear blocks with 4 breakpoints (or less), provided the same performance as the Output-Error model of the second order. This implies that the relationships between the GVS artifacts at the EEG electrodes and the injected GVS current are linear and remain constant over the entire epoch. Our simulation study results also showed that the impedance models between the EEG electrodes and the GVS electrodes remain constant over the entire epoch  and usinWe also showed that, when we apply the proposed method to remove the GVS artifacts, less distortion is introduced in the cleaned EEG signals, compared to the distortion that the other methods  introduce. Furthermore, using the proposed method, we do not need to collect and process all EEG channels as in the ICA-based analysis; therefore it is much faster than the ICA-based methods. This allows us to have a simple experimental setup for collecting EEG signals with less EEG channels for the GVS studies which makes the preparation for the data acquisition session take less time before the subject gets tired, and more myogenic and ocular artifacts are introduced. Compared to the ICA methods, the proposed method is easier to be implemented in a real time system for future applications.


The thalamus modulates the information flow to and from the cortex. The basal ganglia provide the thalamus with inhibitory input, whereas the cortex sends excitatory connections to the thalamus. Thalamocortical relay (TCR) neurons can fire both single spikes and bursts: the latter ones after a low threshold T-type calcium current is activated from a sufficiently hyperpolarized membrane potential. The question what these spikes and bursts code for and how the basal ganglia can influence this coding still remains open.Information processing of TCR neurons was investigated in a computational model and compared with data recorded from TCR neurons in-vitro patched in rat thalamic brain slices. Gaussian noise superimposed on a DC current that put the neuron in different firing regimes (bursting or spiking) was injected into the TCR neuron and several computational tools to measure the reliability, precision and information content of resulting spike trains and coherence between input and output were used to analyze the recorded spike trains. Reverse correlation techniques (Spike-Triggered Average (STA) and Spike-Triggered Covariance (STC)) revealed what input features the neurons were most sensitive to in the different regimes.The results of the analysis show that the two firing regimes of the TCR neurons, bursting mode and spiking mode, represent different coding strategies. At a low membrane potential, the neuron is in bursting mode, in which bursts phase-lock preferentially to low frequencies in the input signal, up to about 20 Hz, and single spikes show little reliability. The STA and filters of the STC analysis show that bursts are a response to long-lasting (up to 200 ms) input signals with a strong negative component. At more depolarized membrane potentials the neuron shows a gradual change from a bursting to a spiking regime. In this second regime single spikes are more precise and more reliable, earlier in time, and phase lock to input frequencies up to 60 Hz. The peaks of the STA become narrower, which reflects a shorter integration time and lower input resistance of the neuron. The STC filters show that spikes are more selective for fast fluctuations in this regime.These results show that even at the single neuron level the thalamus can use different coding strategies depending on whether it is in a more hyperpolarized state, for instance as a consequence of strong basal ganglia input, or in a more depolarized state, induced by strong cortical input. This implies that single neurons can change their encoding regime depending on the background activity of the surrounding network.


To understand the function of protein complexes and their association with biological processes, a lot of studies have been done towards analyzing the protein-protein interaction (PPI) networks. However, the advancement in high-throughput technology has resulted in a humongous amount of data for analysis. Moreover, high level of noise, sparseness, and skewness in degree distribution of PPI networks limits the performance of many clustering algorithms and further analysis of their interactions.In addressing and solving these problems we present a novel random walk based algorithm that converts the incomplete and binary PPI network into a protein-protein topological similarity matrix (PP-TS matrix). We believe that if two proteins share some high-order topological similarities they are likely to be interacting with each other. Using the obtained PP-TS matrix, we constructed and used weighted networks to further study and analyze the interaction among proteins. Specifically, we applied a fully automated community structure finding algorithm (Auto-HQcut) on the obtained weighted network to cluster protein complexes. We then analyzed the protein complexes for significance in biological processes. To help visualize and analyze these protein complexes we also developed an interface that displays the resulting complexes as well as the characteristics associated with each complex.Applying our approach to a yeast protein-protein interaction network, we found that the predicted protein-protein interaction pairs with high topological similarities have more significant biological relevance than the original protein-protein interactions pairs. When we compared our PPI network reconstruction algorithm with other existing algorithms using gene ontology and gene co-expression, our algorithm produced the highest similarity scores. Also, our predicted protein complexes showed higher accuracy measure compared to the other protein complex predictions. Protein-protein interaction (PPI) is the core to many fundamental biological processes. New high-throughput techniques, such as yeast two-hybrid and tandem affinity purification , have vaThe growing PPI database helps both biological and computational scientists to predict gene functions, functional pathways, protein complexes and improve the diagnosis and treatment of diseases -16. HoweIn biological networks, community structures (protein complexes) normally have key biological significance. Identifying and analyzing such communities leads to the understanding of the nature of the system as well as potential novel relationships between communities . One appRecently, we proposed a novel idea to predict protein-protein interactions based on topological similarity between nodes in a given PPI network ,22. BasiIn this work, we extend our work to introduce a novel approach for determining an optimal cutoff to predict protein-protein interactions from the weighted topological similarity matrix. Our idea is that a good cutoff should result in a network that is highly modular. Based on this idea, we also present a parameter-free, recursive algorithm, Auto-HQcut, to identify protein complexes from the PPI network. We also developed a graphical user interface to enable visualization of the weighted network and the predicted protein complexes.To evaluate our approach, we applied the network reconstruction algorithm to a yeast PPI network and examine the biological relevance of the resulting PPI network. The resulting PPI network showed interactions with more functional relevance in comparison to the original PPI network. Comparison with existing methods showed that the network reconstructed by our method has the highest overall quality. Furthermore, applying Auto-HQcut clustering algorithm, we found that the reconstructed network had significantly improved prediction accuracy of protein complexes.For evaluation, we applied our algorithm to a yeast core PPI network obtained from , which cFor clustering, we used the methods based on the optimization of a modularity function (Q). We developed an approach, Auto-HQcut, to automatically determine the best parameter for the algorithm HQcut. We compared the clustering accuracy of each resulting protein complex to the known complexes. The results showed that our approach could be used to deduce the significant biological processes from any PPI network.The performance of the algorithm can be tested by evaluating the pairwise genes with top similarity scores in the PP-TS matrix for functional relevance. It is expected for the gene pairs with high PP-TS scores to be functionally more relevant. To determine the functional relevance between the gene pairs, we used the functional annotations in gene ontology (GO) and gene expression (GE) patterns across diverse conditions.To represent the functional relevance of the top gene pairs in the PP-TS matrix, we calculated the average gene ontology similarity  and gene co-expression  of the gene pairs with the top PP-TS scores, as shown in Figure For GO, we used the gene pairs with top PP-TS score to obtain the same average GO score as the original PPI network to identify more associated gene pairs (26170 compared to 7123) which were not available in the original network. On the other hand, if we keep the same number of edges  , random The average GE similarity scores are shown in Figure As the PP-TS matrix is a weighted and completely connected network, thresholds values are used to convert it to a sparse network. To this end, we chose the cutoff value such that the average GO value of the interacting pairs in the modified network is the same as that of the original PPI network. This resulted in 26170 edges, as mentioned above. We then applied two network clustering algorithms, Qcut and HQcut, to the original and RWS modified networks, and compared the predicted complexes with the MIPS known protein complexes (see Methods). As shown in Figure Again, we compared our algorithm with the four existing methods, ECT, RWR, MDS and GGA. For a fair comparison, we choose a different cutoff for each algorithm so that the predicted PPI networks from different algorithms all have the same number of edges as the RWS modified PPI network. These networks are then subject to protein complex prediction using the HQcut algorithm. As shown in Figure In order to further improve the protein complex prediction accuracy of HQcut, we reconstructed a series of weighted PPI networks with different cutoffs, and run HQcut on each of the network to identify potential protein complexes. We also measured the modularity differences between the selected networks and random generated networks. As shown in Figure Figure To display the result of protein complexes obtained, we developed an interface to show the complexes and their associated properties. The result on Figure Random Walk with Resistance(RWS) [Since the original PPI network is binary, incomplete in nature, and is not good for the clustering algorithms, we used the nce(RWS)  to reconG be an undirected graph, V the set of vertices and E the set of edges. With the basic random walk idea, the probability to move from an initial node v to its neighbor node u in the next step is Pvu = 1/d(v) for  ∈ E, where d(v) is the degree of node v. In RWS algorithm, we introduced two types of resistance ε and β and the probability for a random walker initiated from node v to take the edge  at time point k + 1 can be calculated by Equation (1), whereLet j at time point k + 1 can be calculated asand the probability for the random walker to reach node ε) makes the final convergence status different for each node based on its topology within the network. The second type of resistance(β) is used to control the depth of a random walk and helps to avoid the hub node effect. When applying the RWS algorithm to the original PPI network, we obtain all the topological profiles for each node. These topological profiles are used to calculate the correlation between all pairs of nodes, where correlation value represents the topological similarity between two nodes. The combined topological similarity for all pairs of nodes builds the PP-TS matrix, which can be viewed as a network with weighted edges.The RWS algorithm is a modified Random Walk algorithm which uses two types of resistances. The first type of resistance(Q) of a given network [In clustering the network, we first apply Qcut algorithm  which op network . HoweverQcut algorithm involves two steps: partitioning and refining. In partitioning, the network is recursively divided each time calculating the Q value until no further improvement is found in the Q value. In refining stage, the communities obtained in the partitioning stage are further refined to try to improve the Q value. In order to achieve this, the communities obtained are changed by: either moving the vertex from one community to another, combining two communities to form a new bigger community, or splitting a community to form two small communities. If the change in the original community results into a better Q value, a change is considered, otherwise no change is made to the original community.HQcut algorithm  improvesAs seen from the algorithm description above, HQcut can further optimize the Q value hence increase prediction accuracy of protein complexes. However, it is hard to choose the best cutoff to apply on the PP-TS matrix that would produce the network with the best prediction accuracy of the protein complexes. As a result, we run HQcut to the networks generated from PP-TS matrix with different cutoffs, and measure the statistical significance of the resulting Q value by comparing the selected networks with the random generated networks. By using the modularity difference between these two values, we can tell if the selected network can predict the protein complexes from the random one. At the end, we pick the cutoff which gives the largest modularity difference value to be used to generate the protein complexes.To evaluate the biological relevance of the newly predicted edges, we resort to gene ontology and gene expression, which are widely used in the functional evaluation. To measure the biological functional relevance between any pair of genes, we used the semantic similarity between the GO terms annotated with the proteins, with a popular method ,29, whicA be the list of gene pairs that fall into the same complex in the set of predicted complexes and B be the list of gene pairs in the set of known complexes, the prediction accuracy is measured by |A| denotes the cardinality of the set A.To investigate the protein complexes prediction accuracy improvement, we compared the predicted complexes with the MIPS known protein complexes , which iTo help visualize protein complexes obtained from our approach, we developed an interface also used as a plugin in Cytoscape . The intIn this paper, we developed a random walk based algorithm that converts the PPI network into a protein-protein topological similarity matrix which is then used to construct weighted networks. The key idea is that two proteins sharing some high-order topological similarities are likely to be interacting with each other and be involved in the same biological processes. Using the reconstructed weighted network, we can measure the interactions of all the protein-protein pairs using a real value as opposed to the "connected/non-connected" measure in the original PPI network, which also helps to reduce noise and uncover significant biological interactions that would otherwise be overlooked when analyzing the original PPI network. We then used a parameter free modularity based community finding algorithm (Auto-HQcut) to identify protein complexes from PPI network by optimizing the modularity function. Finally, we used the interface we developed to visualize and analyze the characteristics of the protein complexes obtained. The results showed that the algorithm can find higher modularity protein complexes with better prediction accuracy.In summary, the weighted network reconstructed from PP-TS matrix has much higher biological relevance than the original network and Auto-HQcut significantly improved protein complex prediction accuracy. Since our method improved the protein-protein similarity quality without any additional biological information involved, the algorithm can be easily combined with other approaches to improve the analysis of PPI networks and protein complex prediction.The authors declare that they have no competing interests.CL and JR conceived the research. CL implemented the RWS algorithm and performed the experiments. ST designed and implemented the visualization interface. AJRB provided biological background. CL, JR, ST and AJRB analyzed the data. CL, ST and JR wrote the manuscript. All authors read and approved the final manuscript.


It is well established that individuals age differently. Yet the nature of these inter-individual differences is still largely unknown. For humans, two main hypotheses have been recently formulated: individuals may experience differences in aging rate or aging timing. This issue is central because it directly influences predictions for human lifespan and provides strong insights into the biological determinants of aging. In this article, we propose a model which lets population heterogeneity emerge from an evolutionary algorithm. We find that whether individuals differ in (i) aging rate or (ii) timing leads to different emerging population heterogeneity. Yet, in both cases, the same mortality patterns are observed at the population level. These patterns qualitatively reproduce those of yeasts, flies, worms and humans. Such findings, supported by an extensive parameter exploration, suggest that mortality patterns across species and their potential shapes belong to a limited and robust set of possible curves. In addition, we use our model to shed light on the notion of subpopulations, link population heterogeneity with the experimental results of stress induction experiments and provide predictions about the expected mortality patterns. As biology is moving towards the study of the distribution of individual-based measures, the model and framework we propose here paves the way for evolutionary interpretations of empirical and experimental data linking the individual level to the population level. Aging is a widespread phenomenon across the tree of life. From yeast to humans, mortality changes over age have been widely documented. Interestingly, all individuals are not equal with respect to the aging process: large variability in individual life span has been reported, even in clonal populations. Understanding the nature of these differences is of great interest for medical research. So far, two hypotheses have been proposed: individuals may differ in their aging rate or timing. Here, we show that these two hypotheses can reproduce experimental and empirical mortality patterns as a result of natural selection. We also predict the corresponding population heterogeneity in aging. Many studies define subpopulations ad hoc, the work we present provides insight into a more accurate description of inter-individual differences in aging. Finally, our analysis also predicts the modifications of these mortality patterns under stressful conditions. This exploration reproduces experimental data obtained with heat shocks and permits to foresee new mortality patterns that could be observed with other perturbations. Aging can be generally defined as age-related changes in a set of variables, from growth rate to reproductive effort, which influence the fitness of an organism. Aging is a multiscale process which can be measured at almost every level of the individual organism. Individual metrics of aging include a broad range of processes from damage to DNA and proteins Age-specific mortality is arguably one of the most documented measure of aging. Since Gompertz' seminal work on human data The nature of inter-individual differences in aging is crucial to study and forecast a population's life expectancy, but is as yet still largely unknown. For humans, two main hypotheses have been proposed recently Drosophila melanogaster populations: when facing a diet restriction, a change in aging timing occurs while the response to fluctuations in temperature is a change in aging rate Caenorhabditis elegansBecause of these exponential patterns, differences in aging rates and timing can be visualized respectively as changes in slope and shift in mortality patterns These mortality curves are population measures: they result from the aggregation of individual's aging. Biodemographic studies of aging have shown that specific population heterogeneity can reproduce the main features of qualitatively different mortality patterns, such as late-age mortality plateaus In our model, we include Gompertz aging, that is exponential increase in mortality hazard with age, and implement heterogeneity in aging rates and timing. Population heterogeneity is allowed to evolve over generations under the influence of life-history trade-offs. Following the disposable soma theory In section ‘Transitions in mortality curves’, we discuss predictions concerning population heterogeneity and the corresponding mortality patterns under different assumptions about inter-individual differences. We compare our results to mortality patterns of yeast, flies, worms and humans respectively. In section ‘Influence of mutation rate’, we test the robustness of our results with respect to mutation rates and highlight new features for the distribution of heterogeneity. Finally, in section ‘The notion of subpopulation’, we exploit the predictions of the model to shed light on the notion of subpopulations which is invoked in numerous experimental studies The model we propose describes evolving populations in which individuals age, die and compete for reproduction while they are alive. The offspring they produce fill the next non-overlapping generation until the desired population size is reached. Survival and reproductive success of each individual are connected following the disposable soma theory. The more an individual invests in reproduction, the shorter its lifespan. For each individual, a single parameter For each generation, all the individuals are synchronized, starting with age zero. While an individual Competition for reproduction has been documented in a broad range of species and can take numerous forms, from limited access to habitat Once an individual is chosen for reproduction, it produces one offspring which inherits its The model we propose simulates evolving populations with a fixed size and non-overlapping generations. We start studying individuals at maturity, so that they are able to reproduce since time Building the generation Following the disposable soma theory, we implement a trade-off at the individual level between survival and reproduction. For each individual, the hazard of death between age First, we implement Second, to derive a mortality function for the Heterogeneity in aging Timing Model (HTM), let us now consider linear dependence on The two hypotheses concerning heterogeneity in aging postulate different underlying trade-off mechanisms. In the case of heterogeneity in aging rates, investing in maintenance mechanisms slows down the aging process whereas in the case of heterogeneity in aging timing, the same investment delays the aging process. In Over generations, the evolutionary process reshapes the distribution of After several hundreds of generations (400 in the simulations below), the distribution stabilizes and mortality curves can be estimated. The probability for a whole population to disappear in the course of evolution is strictly positive. Indeed, each individual has a non-zero probability to die at each time step and this could happen before it reproduces at all. Population size can theoretically decrease over generations until extinction. Yet, for a broad range of parameters, this does not occur and a quasi-stationary distribution of strategies is reached. In In this paper, we focus on adult mortality patterns and, as such, we do not explicitly address the heterogeneity arising from developmental processes. We assume that development leads to the expression of the inherited phenotype In this section, we present the results of evolution in different environments under two distinct assumptions, heterogeneity in aging rates (HRM) and heterogeneity in aging timing (HTM), and compare the outcome with empirical and experimental data. In Simulation results with both the HRM and HTM, presented in In The transitions in mortality patterns presented in In both models, population heterogeneity is shaped by natural selection, via a life-history trade-off between survival and reproduction. The transitions observed in the two versions of the model rely on a change of the underlying heterogeneity in In the case of heterogeneity in aging timing, population heterogeneity evolves with the parameter In our models, individuals are competing with each other to access reproduction. Different Mutations occurring on First, we find that the properties of asymmetry in heterogeneity described in Results in sections ‘Transitions in mortality curves’ and ‘Evolution of heterogeneity’ suggest that changes in population heterogeneity have a considerable influence on mortality patterns, providing the transitions in shape previously described. Yet, we find here that if the left-side of the Because it explicitly links the individual level and population measures of aging, the model we propose also allows for study of what is called ‘subpopulations’ in several experimental studies. In the case of heterogeneity in aging rates, the distribution of individual lifespan in evolved populations is bimodal for When the right tail of the A subpopulation also arises in studies focused on stress induction experiments As a case study, we focus here on stress induction experiments in C. elegans. Whether stress induction experiments induce adaptation which modifies population heterogeneity is a burning issue In both figures, the distribution of The results presented in the other columns provide predictions concerning the expected mortality patterns for other population heterogeneity. The interests of this exploration are twofold. First, it allows one to determine whether a given stress induction experiment will modify population heterogeneity. If the same distribution of Studying the evolution of heterogeneity in resource allocation strategies is at the heart of a broad range of experimental studies and today's aging research Moreover, in terms of medical treatments, focusing on aging rate or timing has distinct implications. On the one hand, modifying the rate of aging results in a high potential increase in life expectancy as aging is slowed during the whole life time. On the other hand, such changes in rate would not only extend the period of ‘healthy aging’ but also morbidity. These pros and cons have arguably significant implications in terms of public policy and need to be considered with regards to population heterogeneity. If the rate of aging is a fundamental constant in humans while timing is highly variable, being able to modify aging rate via medical progress will require substantial investment while timing would be presumably more flexible and thus cheaper to modify.Numerous studies in current aging research have found mutations or treatments which dramatically influence aging dynamics In this model, we analyze different possible heterogeneity in aging, only assuming Gompertz dependence on time. Our results show that assumptions about internal life-history trade-offs modulate the shape of population heterogeneity. Yet, in a wide array of scenarios, the same set of mortality and survival curves emerge. Population heterogeneity adapts to the environment according to the individual mortality functions but, at the population level, no qualitative changes can be observed. The transitions observed always go from a ‘kink’ shaped mortality curve to an exponential-U shape-exponential pattern. This robustness of the mortality patterns also echoes experimental observations of mortality and survival in different environments. In many organisms, the modifications observed in mortality patterns as a result of genetic manipulations The systematic exploration of different forms of inter-individual differences in aging also provides predictions about the expected distributions of heterogeneity. Even though the mortality patterns look alike in the different versions of the model, the evolved heterogeneity distributions do not. Current technical advances, such as microfluidics set-ups Previous works have used the Price equation to link population heterogeneity and natural selection Moreover, the shape of these distributions also provides information about the link between the marker observed at the individual level and the measure of interest, such as individual lifespan. Comparing the distributions emerging from our evolutionary algorithm shows that to obtain the exponential-plateau-exponential pattern  with a unimodal distribution of the aging marker, the link between the marker and individual lifespan needs to be strongly non-linear. The exploration we propose here also provides milestones to interpret the individual-based data and their corresponding distributions.Finally, we believe that the model is also flexible enough to allow the exploration of other types of life-history trade-offs, based on the key idea of evolving heterogeneity. As such, it paves the way for future interpretations of coming individual-based observations in evolving biological systems.The simplicity of the framework we propose also enables the formalization of intuitive notions, such as a ‘subpopulation’ and the extensive exploration of mortality functions provides predictions of possible curves for organisms yet to be studied, along with expected distributions of heterogeneity. The robustness of mortality patterns observed suggests that aging is itself a robust process, relying on similar processes across species. We hope that this work paves the way for (i) faster understanding and classification of heterogeneity distributions across species which are not model organisms and (ii) opens up new prospects in terms of understanding the evolution of aging and its robustness. The set-up allows easy changes and explorations, as well as creating space to make the interactions between aging and reproduction more complex. In that sense, it complements previous approaches combining evolutionary theories and heterogeneity, as it provides a framework to explore yet to be explained aging dynamics.We have simulated our model with both a continuous time and a discrete time framework, in both a stochastic and a deterministic manner, in order to ensure the robustness of our results. All models have been implemented in C and this section provides numerical methods for the algorithm used in our simulations.In the stochastic models, the population has a finite size which is usually 500 individuals, unless otherwise mentioned. Modifying this parameter does not alter the conclusion of the models and the same results can be obtained for different population sizes. The mortality curve we observe as a result of the evolutionary algorithm depends on the parameter In the discrete time version of the stochastic model, the age of each individual is incremented by one at each time step and followed by a survival test. This test consists of drawing a random number The continuous time version of the stochastic model follows the same principles. The time of death and reproduction events are drawn following continuous distributions. First, the time of death for each individual is randomly drawn following an exponential distribution which corresponds to its We have also implemented a deterministic model which corresponds to the stochastic models presented in the main text in the case of infinite populations. The purpose is twofold: (i) it ensures the robustness of our results and (ii) it paves the way for an analytical analysis of the evolutionary algorithm we present in this paper. For instance, the formalisation presented in this section allows one to study from a formal standpoint the convergence to a stationary distribution with infinite populations. The deterministic model is defined at the population level and describes the changes in For computational reasons, the possible values of In supplementary material, we also describe the explorations mentioned in the main article  and provFigure S1Exploration of the parameter space A - Mortality pattern for a mortality function (EPS)Click here for additional data file.Figure S2HRM where mutations consist in perturbing the previous value of (EPS)Click here for additional data file.Figure S3In the HRM, adding mild extrinsic mortality ((EPS)Click here for additional data file.Figure S4Transitions with the (EPS)Click here for additional data file.Figure S5Mortality pattern and stationary distribution obtained with sexual reproduction. The horizontal line shows 0.01 to highlight the decrease of the tail close to 1. ((EPS)Click here for additional data file.Figure S6Mortality pattern and stationary distribution obtained with a maturation time of 10. The horizontal line shows 0.01 to highlight the decrease of the tail close to 1. ((EPS)Click here for additional data file.Figure S7In the absence of mutations, the distribution of (EPS)Click here for additional data file.Figure S8For small mutation rates, population heterogeneity remains bimodal in the case of heterogeneity in timing (HTM). Heterogeneity is maintained because of the time-dependent competition, as described in the mathematical model in (EPS)Click here for additional data file.Figure S9The grey area represents the set of valid couple (EPS)Click here for additional data file.Program S1Simulation Code. Program S1 contains a C code allowing to reproduce the mortality curves of the paper.(ZIP)Click here for additional data file.Text S1Exploration of the parameter space and mathematical models. (PDF)Click here for additional data file.


The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos. We expose a remarkable relationship between nonlinear dynamical systems and complex networks by means of the horizontal visibility (HV) algorithm The horizontal visibility (HV) algorithm converts the information stored in a time series into a network, setting the nature of the dynamical system into a different context that requires complex network tools Model: Feigenbaum graphsThe HV graph raph see . Notice A deep-seated feature of the period-doubling cascade is that the order in which the positions of a periodic attractor are visited is universal i.e.We turn next to the period-doubling bifurcation cascade of chaotic bands that takes place as val  see . This isBefore proceeding to interpret these findings via the consideration of renormalization group (RG) arguments, we recall that the Feigenbaum tree shows a rich self-affine structure: for In order to recast previous findings in the context of the renormalization group, let us define an RG operation odes see .a. This Next, under the same RG transformation, the self-affine structure of the family of attractors yields sity see .Finally, let us consider the RG flow inside a given periodic window of initial period Interestingly, and at odds with standard RG applications to  scale-invariant systems, we find that invariance at i.e. the 1st trivial fixed point of the RG flow. Remarkably, these results indicate that the fixed-point structure of the RG flow are obtained via optimization of the entropy for the entire family of networks, supporting a suggested connection between RG theory and the principle of Maximum Entropy Finally, we derive, via optimization of an entropic functional for the Feigenbaum graphs, all the RG flow directions and fixed points directly from the information contained in the degree distribution. Amongst the graph theoretical entropies that have been proposed we employ here the Shannon entropy of the degree distribution i.e. sensitivity to initial conditions.The network entropy In summary, we have shown that the horizontal visibility theory combines power with straightforwardness as a tool for the analytical study of nonlinear dynamics. As an illustration we have established how the families of periodic and chaotic attractor bifurcation cascades of unimodal maps transform into families of networks with scale-invariant limiting forms, whose characterization can be deduced from two basic and universal properties of unimodal maps: ordering of consecutive positions in the attractors and self-affinity. Further, we have demonstrated that these networks and their associated degree distributions comply with renormalization group and maximum entropy principles, filtering out irrelevant variables and finding fixed-point networks which are independent of the map's nonlinearity. The entire Feigenbaum scenario is therefore fully described. The potential of the theory for revealing new information is indicated by the ability of the network entropy to emulate the Lyapunov exponent for both periodic and chaotic attractors. Extensions of this approach to other complex behavior, such as dynamical complexity associated to vanishing Lyapunov exponents, intermittency, quasiperiodic routes to chaos, etc., are still open questions. Finally, observe that in symbolic dynamics


As there have been very few Chinese herbal patent infringement cases, it is still unclear how the Doctrine of Equivalents should be applied to determine the scope of 'equivalents' in Chinese fufang prescriptions. Case law principles from cases in other technical areas such as chemical patents and biological drug patents can be borrowed to ascertain a precise scope of a fufang patent. This article summarizes and discusses several chemical and biopharmaceutical patent cases. In cases where a certain herbal ingredient is substituted by another herb or a combination of herbs, accused infringers are likely to relate herbal drug patents to chemical drug patents with strict interpretation whereas patent owners may take advantage of the liberal application of Doctrine of Equivalence in biopharmaceutical patents by analogizing the complex nature of herbal drugs with biological drugs. Therefore, consideration should be given to the purpose of an ingredient in a patent, the qualities when combined with the other ingredients and the intended function. The scope of equivalents also depends on the stage of the prior art. Moreover, it is desirable to disclose any potential substitutes when drafting the application. Claims should be drafted in such a way that all foreseeable modifications are encompassed for the protection of the patent owner's intellectual property.Herbal ingredients in a Chinese  Many herbal ingredients in a fufang prescription may be replaced by one or several other herbal combinations without failing to produce similar therapeutic effects. For example, Rhizoma Coptidis (Huanglian), Cortex Phellodendri (Huangbai) and Radix Scutellariae Baicalensis (Huangqin) share similar functions and may be replaced with each other for clearing heat, drying dampness, draining fire and relieving toxicity. Ginseng, when taken orally as adaptogen, aphrodisiac and nourishing stimulant, may be substituted by other herbs in the ginseng family, such as Dangshen and Huangqi.In Chinese medicine practice, single herbal ingredient prescriptions are referred to as fufang patent X comprising A, B, C and D, does a composition consisting of A, B, C and P, or a composition consisting of A, B, C, E and F infringe patent X? The answer to this hypothetical question depends on how the court interprets the scope of the claim coverage in a fufang patent.The existence of multiple substitutes often makes patent owners concerned about limiting the scope of their claims to a particular herbal combination. For example, if someone obtains a To ascertain the precise scope of a patent, one must look at 1) the literal scope of a patent claim and 2) the scope of claim coverage under the Doctrine of Equivalents. Claim construction analysis begins with the literal words of the claim, which generally carries their ordinary and customary meanings  liability is nevertheless appropriate to prevent what is in essence a pirating of the patentee's invention' .The principal limitation on the scope of equivalents to which a patentee may be entitled is prosecution-history estoppel. The essence of prosecution-history estoppel is that patent owners may not recapture through litigation any subject matter they previously surrendered during prosecution. Prosecution-history estoppel prevents patent owners from using the doctrine of equivalents during litigation to argue that the scope of a patent claim should be interpreted to include subject matter surrendered in a narrowing amendment or implicitly by argument .Festo VIII, the Federal Circuit's decision in Festo Corp. (Festo IX) [After remand from the Supreme Court construed the Supreme Court's decision in esto IX)  held thaesto IX) . A patenesto IX) .fufang patents, how the Doctrine of Equivalents will be applied to determine the scope of an 'equivalents' is still unclear. The court has received very few Chinese herbal patent infringement cases. As such, one must borrow case law principles from cases in other technical areas. The most related technical area probably is chemical drug patents, where the Doctrine of Equivalents has matured. The next related technical area is biological drug patents. For the purpose of this article, a few chemical and biotechnological patent cases are summarized and discussed help the readers understand how the Doctrine of Equivalents could be applied to Chinese herbal patents.For Chemical drug patents are often interpreted strictly. A charge of infringement would likely be avoided by any addition, elimination of an active ingredient, or a change in their proportions . The couParmlee Pharmaceutical Co. v. Zink, which involves United States Patent No. 2,478,182 issued to William V. Consolazio [One illustrative example is nsolazio . Parmlee'An internally reinforced sodium chloride tablet comprising compressed granules of sodium chloride; and an internally disposed cellular stroma of a thin, permeable, dialyzing film of a material selected from the group consisting of cellulose acetate and cellulose nitrate, the cells of said stroma containing said granules of sodium chloride, whereby the sodium chloride is rendered slowly available when the tablet reaches the gastro-intestinal tract, the solution time of the sodium chloride in said tablet in the gastrointestinal fluids being from 60 to 80 minutes for a ten grain tablet.'The specification of the patent refers to the use of salt tablets to combat the ill effects of heat and excessive sweating. Ingestion of a salt tablet is frequently associated with incidence of epigastric discomfort, nausea and vomiting. A popular theory at the time of the invention was that quick absorption of the salt tablet would shorten the irritation time to the gastro-intestinal tract. The patentee's innovative concept, however, is that a slow absorption rate of salt would eliminate the discomfort in gastro-intestinal tract entirely. The slow-release result was accomplished by coating the salt tablet with a cellular stroma of a film made from cellulose acetate or cellulose nitrate.The plaintiff's product became a commercial success. The accused tablet conformed in general construction to the patentee's tablet and accomplished the same result. The film employed in the accused infringer's tablet, however, was not of 'the group consisting of cellulose acetate and cellulose nitrate'. The film was, instead, 'shellac'.every material which might be used to form the particular structure, ie a thin permeable dialyzing film disclosed in the patent.The accused product is not a literal infringement of the '182 patent, because the patent's claim was limited to tablets coated with cellulose acetate or cellulose nitrate'. The plaintiffs argued that the accused tablets were equivalents and reasoned that Consolazio's approach to controlled release was radically different from that of the past art, and accordingly, the patent was entitled to a broad range of equivalents which would embrace The defendants argued that numerous prior art had taught what materials could be applied to make the coating and the use of the word 'consisting' in the claim and its reference to cellulose acetate and cellulose nitrate was restrictive and operated to exclude all substances not belonging to that group. The Federal Circuit agreed with the defendants that while the coatings performed the same function, so many different coatings were known prior to the patent's filing that it would be inequitable to expand the scope of the claim beyond the two listed substances. In the light of the prior art, the court held that 'the proper range of equivalents for this patent is a narrow one and ... is of insufficient breadth to include a substance' other than the two recited in the claim .Tanabe Seiyaku Co., Ltd. v. U.S. International Trade Commission, the question of whether acetone (CH3C( = O)CH3) was equivalent to butanone (CH3C( = O)CH2CH3) as a solvent for a chemical reaction was argued before the court [In another case, he court . The plaThe chemical reaction recited in claim 1 is known as an 'N-alkylation' reaction. The five base-solvent combinations disclosed in the '035 patent and recited in claim 1 are listed in Table The defendant presented evidence that duplicating examples from the patent with butanone instead of acetone often gave poor results, while in one case, the result with butanone was better. A good deal of experimentation was performed by the defendant while optimizing a scaled-up reaction, from which the court inferred that the defendant had designed around, rather than copied, the patented method, and that butanone and acetone were not truly interchangeable.While chemical drugs are small molecules mostly produced by chemical synthesis, biological drugs are macromolecules consisting of thousands of atoms. They are made by living cells  whose DNA has been modified by introduction of the gene of interest to synthesize the active component. The complex structures of biological drugs are often composed of multiple long chains of amino acids, derivatized by sugar moieties and folded by complex mechanisms.Due to its complexity, biological drugs cannot be fully copied. No two cell lines, developed independently, can be considered identical. A multiplicity of DNA sequences encodes the same protein. One or several amino acid additions, substitutions or deletions can be made in one protein sequences with retention of the desired activity. As a consequence, the term 'biosimilar' is used to acknowledge the fact that while biosimilar products are similar to the original product, they are not exactly the same.ie one that produces no alteration in the amino acids sequence of the expressed protein, will always be an infringement under the Doctrine of Equivalents [Compared to the chemical drug patents, claims for biological drug patents are interpreted more broadly. The courts have applied the Doctrine of Equivalents more liberally . When deivalents .Genentech, Inc. v. The Wellcome Foundation, Ltd., Genentech owned three patents, namely a patent directed to a natural protein extracted from certain human cancer cells, a patent directed to the materials needed to produce the natural protein through recombinant DNA technology and the microorganism or cell culture capable of expressing the protein. The Wellcome Foundation produced a protein through recombinant DNA technology. The accused version of fPA differred by one amino acid in a kringle region, with a half-life ten times of natural t-PA. Nevertheless, the accused protein exhibited the same clot dissolving activity of tPA. A jury at the trial court held that the accused version infringed Genentch's patent under the Doctrine of Equivalents, while concluding that they did not fall literally within the scope of Genentech's patent. The district court holding was reversed by the Federal Circuit [On the other hand, if the difference between the patented compound and the accused compound is substantial and significant, a finding of equivalents might not sustain . In Gene Circuit . The cou Circuit .Festo IX [Nevertheless, given the impact of the Festo IX  owners oFesto IX . TherefoSchwarz Pharma v. Paddock, Schwarz Pharma owns U.S. Patent 4,743,450 ('the '450 patent') [For example, in patent') . The '45patent') .Paddock filed abbreviated new drug application (ANDA) for generic moexipril formulation, using magnesium oxide (MgO) as a stabilizer. Schwarz Pharma sued Paddock for patent infringement, arguing that MgO is a foreseeable equivalent of magnesium carbonate described in patent. The District Court of Minnesota found non-infringement based on prosecution history estoppel. Originally, the independent Claim 1 recited 'an alkali or alkaline earth-metal salt'; however, it was amended to instead recite 'an alkali or alkaline earth metal carbonate' following the rejection. The court held that the change in claim language was a narrowing amendment and presumptively surrendered all metal containing stabilizers and alkali or alkaline earth metal salts except alkali and alkaline earth metal carbonates. The court also held that Schwarz had failed to rebut the presumption of surrender because magnesium oxide was a foreseeable equivalent of magnesium carbonate and because there was no objectively apparent reason for the narrowing amendment not directly related to the use of magnesium oxide. The court thus concluded that the Paddock drug did not infringe because Schwarz was estopped from claiming that the magnesium oxide used by Paddock was the equivalent of an alkali or alkaline earth metal carbonate. The decision was later confirmed by the Federal Circuit.As very few herbal patent infringement cases have been decided by the court, how the Doctrine of Equivalents will be applied in herbal patents is unclear. Both patent owners and the accused infringers would have to refer to case law principles from chemical and/or biological patent cases.fufang herbal drug often includes over a dozen herbs. The complexity of an herbal drug is further enhanced by the production process. As such, every herbal drug can be considered unique and this is very similar to biological drugs. Substituting or changing the amount of a minor herbal component in a fufang drug will lead to a 'different' composition but the change may or may not cause difference in the overall therapeutic effect. Therefore, 'similarities' rather than 'identity' should be compared. Unless the 'similarities' of the herbal drugs are changed, the substitute should be considered as an 'equivalent' and within the scope of the claims.As the active ingredients identified in herbal medicines are mostly small molecules, it is reasonable that herbal patents should follow the general rule for chemical drugs. The accused infringers are likely to argue that the substituted herbal ingredient adds new chemical compounds into the drug composition, thereby performing different functions. Herbal drug patent owners, nevertheless, may argue that herbal drugs are very different from chemical drugs due to the complex nature of herbal drugs. Herbal drugs often contain multiple active ingredients; furthermore, a fufang drug may alter its therapeutic effects and potential side effects. The original patented fufang composition and the substituted fufang should not be considered as 'equivalents'. However, minor distinctions  should be recognized under these circumstances. Another important factor is whether persons reasonably skilled in the art would have known of the interchangeability of the substituting ingredient [For example, substitution of a principal herbal ingredient in a gredient .The best time to protect the patent owner's intellectual property is at the time of drafting the application. While it is impossible to cover all possible equivalents, it is advisable to disclose any potential substitutes. Claims should be drafted to literally encompass all foreseeable modifications. The specification should also disclose any 'functional equivalents' at the time of drafting. The applicant must define exactly which variations may be included, possibly based on some experimental result and clinical data .In drafting a patent application for a Chinese herbal drug, one should give consideration to the intended purpose of an ingredient, the quality when combined with the other ingredients and the intended function. What constitutes an equivalent must be determined against the context of the patent, the prior art and the particular circumstances of the case. The range of equivalents is broader in an uncrowded art and narrower in a crowded one.The authors declare that they have no competing interests.AWKC conceived the study and revised the manuscript. XW and JT researched on the subject and drafted the manuscript. All authors read and approved the final version of the manuscript.