Roger Rodriguez

Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks.

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations—the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs withsuperimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.

Mean instantaneous firing frequency is always higher than the firing rate

Frequency coding is considered one of the most common coding strategies employed by neural systems. This fact leads, in experiments as well as in theoretical studies, to construction of so-called transfer functions, where the output firing frequency is plotted against the input intensity. The term firing frequency can be understood differently in different contexts. Basically, it means that the number of spikes over an interval of preselected length is counted and then divided by the length of the interval, but due to the obvious limitations, the length of observation cannot be arbitrarily long. Then firing frequency is defined as reciprocal to the mean interspike interval. In parallel, an instantaneous firing frequency can be defined as reciprocal to the length of current interspike interval, and by taking a mean of these, the definition can be extended to introduce the mean instantaneous firing frequency. All of these definitions of firing frequency are compared in an effort to contribute to a better understanding of the input-output properties of a neuron.

Determination of firing times for the stochastic Fitzhugh-Nagumo neuronal model

We present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models. We apply the theory of first exit times for Markov processes to the Fitzhugh-Nagumo system with a constant mean gaussian white noise input, representing stochastic excitation and inhibition. Partial differential equations are obtained for the moments of the time to first spike. The observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one-dimensional approximation. For the moments of the time to reach threshold, this leads to ordinary differential equations that may be easily solved. Several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter. For ranges of the parameters appropriate for these asymptotic methods, the perturbation solutions are used to establish the validity of the one-dimensional approximation for both small and large values of the noise parameter. Additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one-dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations. Such agreement extends to intermediate values of the noise parameter. For the mean time to threshold, we find maxima at small noise values that constitute a form of stochastic resonance. We also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean.

Noisy spiking neurons and networks: useful approximations for firing probabilities and global behavior

Electrophysiological properties of spiking neurons receiving complex stimuli perturbed by noise are investigated. A semi-analytical estimate of firing probabilities and subthreshold behavior of the stochastic system can be made in terms of the solution of a purely deterministic system. The method comes from an approximation for the distribution function and moments of the underlying non linear multidimensional diffusion process. This so called moment method works for general conductance-based systems and an application is presented for the Hodgkin-Huxley neuronal model. Statistical properties obtained from the moment method are compared with direct numerical integration of the stochastic system. The firing probability due to external noise is derived as a closed formula. Results are given for different forms of the deterministic component of the stimulus. A generalization to neural networks of conductance-based systems with internal currents perturbed by noise can be obtained using the same approach. In the case of fully connected networks, a mean field population equation is derived which may be compared to Kuramotos master equation for weakly coupled neural oscillators.

The spatial properties of a model neuron increase its coding range

Abstract. The coding properties of one-compartment and two-compartment model neurons are compared. The membrane depolarization in both models is described as a deterministic leaky integrator. Interspike intervals are identified with the periods between reset of the depolarization after firing and consecutive crossing of a fixed firing threshold. The two-point model has an input in the dendritic compartment and an output in the trigger-zone compartment. It is shown that the sensitivity threshold for the two-point model is shifted to the larger values of the input intensity with respect to the sensitivity threshold of its single-point counterpart. Further, its coding range is substantially larger than the coding range of the single-point model.

Analytical determination of firing times in stochastic nonlinear neural models

Abstract We present for the first time an analytical approach for determining the time of firing of nonlinear stochastic neuronal models. The theory of Markov processes is applied to the Fitzhugh–Nagumo system with a constant mean, Gaussian white noise input, representing stochastic excitation and inhibition. An equation obtained for the mean of the time to first spike is solved by means of an accurate one-dimensional approximation. Verification of the method is obtained through the excellent agreement between the moments of the firing time found by numerical solution of differential equations and those obtained by simulation of the solutions of the model stochastic differential equations. We also find a maximum at small noise values which constitutes a form of stochastic resonance.

Coupled Hodgkin Huxley Neurons with Stochastic Synaptic Inputs

The fluctuations of synaptic transmissions in neuronal systems are described, at the scale of refraction time, as inhomogeneous compound jump processes. Active properties of membrane potentials are also introduced and analytical estimates of useful probabilistic quantities can be deduced for biologically plausible dimensional reductions of Hodgkin Huxley type neuron models.

A simple stochastic model of spatially complex neurons

A method for studying the coding properties of a multicompartmental integrate-and-fire neuron of arbitrary geometry is presented. Depolarization at each compartment evolves like a leaky integrator with an after-firing reset imposed only at the trigger zone. The frequency of firing at the steady-state regime is related to the properties of the multidimensional input. The decreasing variability of subthreshold depolarization from the dendritic tree to the trigger zone is shown for an input that is corrupted by a white noise. The role of a Poissonian noise is also investigated. The proposed method gives an estimate of the mean interspike interval that can be used to study the input output transfer function of the system. Both types of the stochastic inputs result in broadening the transfer function with respect to the deterministic case.

Vesicular mechanisms and estimates of firing probability in a network of spiking neurons

Abstract A neural network with mutual excitatory connections and external stimulation is investigated. The units of the network are Morris–Lecar neurons. The synaptic transmission is described at the vesicular level. Random number of activated vesicles at synaptic contacts and random quanta of released transmitter are considered. These fluctuations are applied in a form of inhomogeneous Poisson processes, at the time scale of the spike duration. The parameters of these processes depend on the presynaptic spiking activity and on the strength of afferent connections. It is shown how synchronization of the activity in the network appears. A statistical analysis of spiking times is performed, showing smooth mean behavior of response frequencies. A diffusion approximation of the network Poissonian process is derived from which an analytical formula for firing probability is calculated.

Two-Compartment Stochastic Model of a Neuron with Periodic Input

A neuronal model made of two interconnected parts—a dendrite and a trigger zone—is considered under the action of white noise and periodic inputs acting on the dendritic compartment. It is shown analytically how the variability of the depolarization potential is decreased from the dendritic to the trigger zone for subthreshold behavior. These filtering properties of the model are also shown when a reset mechanism is included at the trigger zone compartment.