Dynamic Finite Size Effects in Spiking Neural Networks

We investigate the dynamics of a deterministic finite-sized network of synaptically coupled spiking neurons and present a formalism for computing the network statistics in a perturbative expansion. The small parameter for the expansion is the inverse number of neurons in the network. The network dynamics are fully characterized by a neuron population density that obeys a conservation law analogous to the Klimontovich equation in the kinetic theory of plasmas. The Klimontovich equation does not possess well-behaved solutions but can be recast in terms of a coupled system of well-behaved moment equations, known as a moment hierarchy. The moment hierarchy is impossible to solve but in the mean field limit of an infinite number of neurons, it reduces to a single well-behaved conservation law for the mean neuron density. For a large but finite system, the moment hierarchy can be truncated perturbatively with the inverse system size as a small parameter but the resulting set of reduced moment equations that are still very difficult to solve. However, the entire moment hierarchy can also be re-expressed in terms of a functional probability distribution of the neuron density. The moments can then be computed perturbatively using methods from statistical field theory. Here we derive the complete mean field theory and the lowest order second moment corrections for physiologically relevant quantities. Although we focus on finite-size corrections, our method can be used to compute perturbative expansions in any parameter.     

Memory Capacity of Networks with Stochastic Binary Synapses

In standard attractor neural network models, specific patterns of activity are stored in the synaptic matrix, so that they become fixed point attractors of the network dynamics. The storage capacity of such networks has been quantified in two ways: the maximal number of patterns that can be stored, and the stored information measured in bits per synapse. In this paper, we compute both quantities in fully connected networks of N binary neurons with binary synapses, storing patterns with coding level , in the large and sparse coding limits (). We also derive finite-size corrections that accurately reproduce the results of simulations in networks of tens of thousands of neurons. These methods are applied to three different scenarios: (1) the classic Willshaw model, (2) networks with stochastic learning in which patterns are shown only once (one shot learning), (3) networks with stochastic learning in which patterns are shown multiple times. The storage capacities are optimized over network parameters, which allows us to compare the performance of the different models. We show that finite-size effects strongly reduce the capacity, even for networks of realistic sizes. We discuss the implications of these results for memory storage in the hippocampus and cerebral cortex.     

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this “paradox,” a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.     

Stochastic models of kleptoparasitism

In this paper, we consider a model of kleptoparasitism amongst a small group of individuals, where the state of the population is described by the distribution of its individuals over three specific types of behaviour (handling, searching for or fighting over, food). The model used is based upon earlier work which considered an equivalent deterministic model relating to large, effectively infinite, populations. We find explicit equations for the probability of the population being in each state. For any reasonably sized population, the number of possible states, and hence the number of equations, is large. These equations are used to find a set of equations for the means, variances, covariances and higher moments for the number of individuals performing each type of behaviour. Given the fixed population size, there are five moments of order one or two (two means, two variances and a covariance). A normal approximation is used to find a set of equations for these five principal moments. The results of our model are then analysed numerically, with the exact solutions, the normal approximation and the deterministic infinite population model compared. It is found that the original deterministic models approximate the stochastic model well in most situations, but that the normal approximations are better, proving to be good approximations to the exact distribution, which can greatly reduce computing time.     

How noisy adaptation of neurons shapes interspike interval histograms and correlations

Channel noise is the dominant intrinsic noise source of neurons causing variability in the timing of action potentials and interspike intervals (ISI). Slow adaptation currents are observed in many cells and strongly shape response properties of neurons. These currents are mediated by finite populations of ionic channels and may thus carry a substantial noise component. Here we study the effect of such adaptation noise on the ISI statistics of an integrate-and-fire model neuron by means of analytical techniques and extensive numerical simulations. We contrast this stochastic adaptation with the commonly studied case of a fast fluctuating current noise and a deterministic adaptation current (corresponding to an infinite population of adaptation channels). We derive analytical approximations for the ISI density and ISI serial correlation coefficient for both cases. For fast fluctuations and deterministic adaptation, the ISI density is well approximated by an inverse Gaussian (IG) and the ISI correlations are negative. In marked contrast, for stochastic adaptation, the density is more peaked and has a heavier tail than an IG density and the serial correlations are positive. A numerical study of the mixed case where both fast fluctuations and adaptation channel noise are present reveals a smooth transition between the analytically tractable limiting cases. Our conclusions are furthermore supported by numerical simulations of a biophysically more realistic Hodgkin-Huxley type model. Our results could be used to infer the dominant source of noise in neurons from their ISI statistics.     

Dynamics of Competition between Subnetworks of Spiking Neuronal Networks in the Balanced State

We explore and analyze the nonlinear switching dynamics of neuronal networks with non-homogeneous connectivity. The general significance of such transient dynamics for brain function is unclear; however, for instance decision-making processes in perception and cognition have been implicated with it. The network under study here is comprised of three subnetworks of either excitatory or inhibitory leaky integrate-and-fire neurons, of which two are of the same type. The synaptic weights are arranged to establish and maintain a balance between excitation and inhibition in case of a constant external drive. Each subnetwork is randomly connected, where all neurons belonging to a particular population have the same in-degree and the same out-degree. Neurons in different subnetworks are also randomly connected with the same probability; however, depending on the type of the pre-synaptic neuron, the synaptic weight is scaled by a factor. We observed that for a certain range of the “within” versus “between” connection weights (bifurcation parameter), the network activation spontaneously switches between the two sub-networks of the same type. This kind of dynamics has been termed “winnerless competition”, which also has a random component here. In our model, this phenomenon is well described by a set of coupled stochastic differential equations of Lotka-Volterra type that imply a competition between the subnetworks. The associated mean-field model shows the same dynamical behavior as observed in simulations of large networks comprising thousands of spiking neurons. The deterministic phase portrait is characterized by two attractors and a saddle node, its stochastic component is essentially given by the multiplicative inherent noise of the system. We find that the dwell time distribution of the active states is exponential, indicating that the noise drives the system randomly from one attractor to the other. A similar model for a larger number of populations might suggest a general approach to study the dynamics of interacting populations of spiking networks.     

Single-Crossover Dynamics: Finite versus Infinite Populations

Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarize and adapt a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If there is only recombination and every pair of recombined offspring replaces their pair of parents (i.e., there is no resampling), then the expected type frequencies in the finite population, of arbitrary size, equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.     

Epidemics with general generation interval distributions

We study the spread of susceptible-infected-recovered (SIR) infectious diseases where an individual's infectiousness and probability of recovery depend on his/her “age” of infection. We focus first on early outbreak stages when stochastic effects dominate and show that epidemics tend to happen faster than deterministic calculations predict. If an outbreak is sufficiently large, stochastic effects are negligible and we modify the standard ordinary differential equation (ODE) model to accommodate age-of-infection effects. We avoid the use of partial differential equations which typically appear in related models. We introduce a “memoryless” ODE system which approximates the true solutions. Finally, we analyze the transition from the stochastic to the deterministic phase.     

The Correlation Structure of Local Neuronal Networks Intrinsically Results from Recurrent Dynamics

Correlated neuronal activity is a natural consequence of network connectivity and shared inputs to pairs of neurons, but the task-dependent modulation of correlations in relation to behavior also hints at a functional role. Correlations influence the gain of postsynaptic neurons, the amount of information encoded in the population activity and decoded by readout neurons, and synaptic plasticity. Further, it affects the power and spatial reach of extracellular signals like the local-field potential. A theory of correlated neuronal activity accounting for recurrent connectivity as well as fluctuating external sources is currently lacking. In particular, it is unclear how the recently found mechanism of active decorrelation by negative feedback on the population level affects the network response to externally applied correlated stimuli. Here, we present such an extension of the theory of correlations in stochastic binary networks. We show that (1) for homogeneous external input, the structure of correlations is mainly determined by the local recurrent connectivity, (2) homogeneous external inputs provide an additive, unspecific contribution to the correlations, (3) inhibitory feedback effectively decorrelates neuronal activity, even if neurons receive identical external inputs, and (4) identical synaptic input statistics to excitatory and to inhibitory cells increases intrinsically generated fluctuations and pairwise correlations. We further demonstrate how the accuracy of mean-field predictions can be improved by self-consistently including correlations. As a byproduct, we show that the cancellation of correlations between the summed inputs to pairs of neurons does not originate from the fast tracking of external input, but from the suppression of fluctuations on the population level by the local network. This suppression is a necessary constraint, but not sufficient to determine the structure of correlations; specifically, the structure observed at finite network size differs from the prediction based on perfect tracking, even though perfect tracking implies suppression of population fluctuations.     

Collective behavior of large-scale neural networks with GPU acceleration

In this paper, the collective behaviors of a small-world neuronal network motivated by the anatomy of a mammalian cortex based on both Izhikevich model and Rulkov model are studied. The Izhikevich model can not only reproduce the rich behaviors of biological neurons but also has only two equations and one nonlinear term. Rulkov model is in the form of difference equations that generate a sequence of membrane potential samples in discrete moments of time to improve computational efficiency. These two models are suitable for the construction of large scale neural networks. By varying some key parameters, such as the connection probability and the number of nearest neighbor of each node, the coupled neurons will exhibit types of temporal and spatial characteristics. It is demonstrated that the implementation of GPU can achieve more and more acceleration than CPU with the increasing of neuron number and iterations. These two small-world network models and GPU acceleration give us a new opportunity to reproduce the real biological network containing a large number of neurons.     

Recurrent epidemics in small world networks

The effect of spatial correlations on the spread of infectious diseases was investigated using a stochastic susceptible-infective-recovered (SIR) model on complex networks. It was found that in addition to the reduction of the effective transmission rate, through the screening of infectives, spatial correlations have another major effect through the enhancement of stochastic fluctuations, which may become considerably larger than in the homogeneously mixed stochastic model. As a consequence, in finite spatially structured populations significant differences from the solutions of deterministic models are to be expected, since sizes even larger than those found for homogeneously mixed stochastic models are required for the effects of fluctuations to be negligible. Furthermore, time series of the (unforced) model provide patterns of recurrent epidemics with slightly irregular periods and realistic amplitudes, suggesting that stochastic models together with complex networks of contacts may be sufficient to describe the long-term dynamics of some diseases. The spatial effects were analysed quantitatively by modelling measles and pertussis, using a susceptible-exposed-infective-recovered (SEIR) model. Both the period and the spatial coherence of the epidemic peaks of pertussis are well described by the unforced model for realistic values of the parameters.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Macroscopic Equations Governing Noisy Spiking Neuronal Populations with Linear Synapses

Deriving tractable reduced equations of biological neural networks capturing the macroscopic dynamics of sub-populations of neurons has been a longstanding problem in computational neuroscience. In this paper, we propose a reduction of large-scale multi-population stochastic networks based on the mean-field theory. We derive, for a wide class of spiking neuron models, a system of differential equations of the type of the usual Wilson-Cowan systems describing the macroscopic activity of populations, under the assumption that synaptic integration is linear with random coefficients. Our reduction involves one unknown function, the effective non-linearity of the network of populations, which can be analytically determined in simple cases, and numerically computed in general. This function depends on the underlying properties of the cells, and in particular the noise level. Appropriate parameters and functions involved in the reduction are given for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models. Simulations of the reduced model show a precise agreement with the macroscopic dynamics of the networks for the first two models.     

Stochastic Oscillations Induced by Intrinsic Fluctuations in a Self-Repressing Gene

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.     

Emergent oscillations in networks of stochastic spiking neurons

Networks of neurons produce diverse patterns of oscillations, arising from the network's global properties, the propensity of individual neurons to oscillate, or a mixture of the two. Here we describe noisy limit cycles and quasi-cycles, two related mechanisms underlying emergent oscillations in neuronal networks whose individual components, stochastic spiking neurons, do not themselves oscillate. Both mechanisms are shown to produce gamma band oscillations at the population level while individual neurons fire at a rate much lower than the population frequency. Spike trains in a network undergoing noisy limit cycles display a preferred period which is not found in the case of quasi-cycles, due to the even faster decay of phase information in quasi-cycles. These oscillations persist in sparsely connected networks, and variation of the network's connectivity results in variation of the oscillation frequency. A network of such neurons behaves as a stochastic perturbation of the deterministic Wilson-Cowan equations, and the network undergoes noisy limit cycles or quasi-cycles depending on whether these have limit cycles or a weakly stable focus. These mechanisms provide a new perspective on the emergence of rhythmic firing in neural networks, showing the coexistence of population-level oscillations with very irregular individual spike trains in a simple and general framework.     

Mathematical analysis of a model for the renal concentrating mechanism

A system of ordinary differential equations, designed to model the counterflow system in the renal medulla, is studied. An existence theorem for solutions of the model equations is obtained. An exact solution of the system is obtained in the limiting case of infinite water permeability. If there is diffusion in the core, evaluation of the exact solution leads to multiple stable solutions of the model equations. One solution has a large concentration ratio, which tends to a finite asymptotic limit as the pump strength tends to infinity.     

Size-Independent Differences between the Mean of Discrete Stochastic Systems and the Corresponding Continuous Deterministic Systems

In this paper, it is shown that for a class of reaction networks, the discrete stochastic nature of the reacting species and reactions results in qualitative and quantitative differences between the mean of exact stochastic simulations and the prediction of the corresponding deterministic system. The differences are independent of the number of molecules of each species in the system under consideration. These reaction networks are open systems of chemical reactions with no zero-order reaction rates. They are characterized by at least two stationary points, one of which is a nonzero stable point, and one unstable trivial solution (stability based on a linear stability analysis of the deterministic system). Starting from a nonzero initial condition, the deterministic system never reaches the zero stationary point due to its unstable nature. In contrast, the result presented here proves that this zero-state is a stable stationary state for the discrete stochastic system, and other finite states have zero probability of existence at large times. This result generalizes previous theoretical studies and simulations of specific systems and provides a theoretical basis for analyzing a class of systems that exhibit such inconsistent behavior. This result has implications in the simulation of infection, apoptosis, and population kinetics, as it can be shown that for certain models the stochastic simulations will always yield different predictions for the mean behavior than the deterministic simulations.