A Fokker-Planck formalism for diffusion with finite increments and absorbing boundaries

Gaussian white noise is frequently used to model fluctuations in physical systems. In Fokker-Planck theory, this leads to a vanishing probability density near the absorbing boundary of threshold models. Here we derive the boundary condition for the stationary density of a first-order stochastic differential equation for additive finite-grained Poisson noise and show that the response properties of threshold units are qualitatively altered. Applied to the integrate-and-fire neuron model, the response turns out to be instantaneous rather than exhibiting low-pass characteristics, highly non-linear, and asymmetric for excitation and inhibition. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled systems of threshold units.Comment: Consists of two parts: main article (3 figures) plus supplementary text (3 extra figures

Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes places, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations, or non-local partial differential equations resembling the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiment also indicate that the McKean-Vlasov-Fokker- Planck equations may be a good way to understand the mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure

Emergence of Synchronous Oscillations in Neural Networks Excited by Noise

The presence of noise in non linear dynamical systems can play a constructive role, increasing the degree of order and coherence or evoking improvements in the performance of the system. An example of this positive influence in a biological system is the impulse transmission in neurons and the synchronization of a neural network. Integrating numerically the Fokker-Planck equation we show a self-induced synchronized oscillation. Such an oscillatory state appears in a neural network coupled with a feedback term, when this system is excited by noise and the noise strength is within a certain range.Comment: 12 pages, 18 figure

Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance

The subthresholdmembranevoltage of a neuron in active cortical tissue is a fluctuating quantity with a distribution that reflects the firing statistics of the presynaptic population. It was recently found that conductancebased synaptic drive can lead to distributions with a significant skew. Here it is demonstrated that the underlying shot noise caused by Poissonian spike arrival also skews the membrane distribution, but in the opposite sense. Using a perturbative method, we analyze the effects of shot noise on the distribution of synaptic conductances and calculate the consequent voltage distribution. To first order in the perturbation theory, the voltage distribution is a gaussian modulated by a prefactor that captures the skew. The gaussian component is identical to distributions derived using current-based models with an effective membrane time constant. The well-known effective-time-constant approximation can therefore be identified as the leading-order solution to the full conductance-based model. The higher-order modulatory prefactor containing the skew comprises terms due to both shot noise and conductance fluctuations. The diffusion approximation misses these shot-noise effects implying that analytical approaches such as the Fokker-Planck equation or simulation with filtered white noise cannot be used to improve on the gaussian approximation. It is further demonstrated that quantities used for fitting theory to experiment, such as the voltage mean and variance, are robust against these non-Gaussian effects. The effective-time-constant approximation is therefore relevant to experiment and provides a simple analytic base on which other pertinent biological details may be added

Fast global oscillations in networks of integrate-and-fire neurons with low firing rates

We study analytically the dynamics of a network of sparsely connected inhibitory integrate-and-fire neurons in a regime where individual neurons emit spikes irregularly and at a low rate. In the limit when the number of neurons N tends to infinity,the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized. The activity becomes oscillatory when the inhibitory feedback is strong enough. The period of the global oscillation is found to be mainly controlled by synaptic times, but depends also on the characteristics of the external input. In large but finite networks, the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold. Their characteristics are determined as functions of systems parameters, in these two different regimes. The results are found to be in good agreement with numerical simulations.Comment: 45 pages, 11 figures, to be published in Neural Computatio

Population dynamics of interacting spiking neurons

A dynamical equation is derived for the spike emission rate nu(t) of a homogeneous network of integrate-and-fire (IF) neurons in a mean-field theoretical framework, where the activity of the single cell depends both on the mean afferent current (the "field") and on its fluctuations. Finite-size effects are taken into account, by a stochastic extension of the dynamical equation for the nu; their effect on the collective activity is studied in detail. Conditions for the local stability of the collective activity are shown to be naturally and simply expressed in terms of (the slope of) the single neuron, static, current-to-rate transfer function. In the framework of the local analysis, we studied the spectral properties of the time-dependent collective activity of the finite network in an asynchronous state; finite-size fluctuations act as an ongoing self-stimulation, which probes the spectral structure of the system on a wide frequency range. The power spectrum of nu exhibits modes ranging from very high frequency (depending on spike transmission delays), which are responsible for instability, to oscillations at a few Hz, direct expression of the diffusion process describing the population dynamics. The latter "diffusion" slow modes do not contribute to the stability conditions. Their characteristic times govern the transient response of the network; these reaction times also exhibit a simple dependence on the slope of the neuron transfer function. We speculate on the possible relevance of our results for the change in the characteristic response time of a neural population during the learning process which shapes the synaptic couplings, thereby affecting the slope of the transfer function. There is remarkable agreement of the theoretical predictions with simulations of a network of IF neurons with a constant leakage term for the membrane potential

Impact of membrane bistability on dynamical response of neuronal populations

Neurons in many brain areas can develop pronounced depolarized state of membrane potential (up state) in addition to the normal hyperpolarized down state near the resting potential. The influence of the up state on signal encoding, however, is not well investigated. Here we construct a one-dimensional bistable neuron model and calculate the linear response to noisy oscillatory inputs analytically. We find that with the appearance of an up state, the transmission function is enhanced by the emergence of a local maximum at some optimal frequency and the phase lag relative to the input signal is reduced. We characterize the dependence of the enhancement of frequency response on intrinsic dynamics and on the occupancy of the up state