Neural network firing-rate models on integral form

Abstract

Firing-rate models describing neural-network activity can be formulated in terms of differential equations for the synaptic drive from neurons. Such models are typically derived from more general models based on Volterra integral equations assuming exponentially decaying temporal coupling kernels describing the coupling of pre- and postsynaptic activities. Here we study models with other choices of temporal coupling kernels. In particular, we investigate the stability properties of constant solutions of two-population Volterra models by studying the equilibrium solutions of the corresponding autonomous dynamical systems, derived using the linear chain trick, by means of the Routh-Hurwitz criterion. In the four investigated synaptic-drive models with identical equilibrium points we find that the choice of temporal coupling kernels significantly affects the equilibrium-point stability properties. A model with an alpha-function replacing the standard exponentially decaying function in the inhibitory coupling kernel is in most of our examples found to be most prone to instability, while the opposite situation with an alpha-function describing the excitatory kernel is found to be least prone to instability. The standard model with exponentially decaying coupling kernels is typically found to be an intermediate case. We further find that stability is promoted by increasing the weight of self-inhibition or shortening the time constant of the inhibition.

Estimation of Thalamocortical and Intracortical Network Models from Joint Thalamic Single-Electrode and Cortical Laminar-Electrode Recordings in the Rat Barrel System

“…For mathematical convenience and conceptual simplicity we choose to formulate the population firing-rate model as Volterra integral equations [5],[14]. In our so called full thalamocortical model where all the abovementioned feedforward and recurrent connections are included, we then have …”

“…We here assume exponentially decaying temporal coupling kernels (Eq. 20) which allows for a mapping of the integral equation to a set of differential equations [5],[14].…”

“…The models are formulated as nonlinear Volterra integral equations with exponentially decaying coupling kernels allowing for a mapping of the systems to sets of differential equations, the more common mathematical representation of firing-rate models [5],[14].…”

Rate dynamics of leaky integrate-and-fire neurons with strong synapses

“…For an ensemble of unconnected LIF neurons, we demonstrated that a simple 1st-order model yields accurate predictions of the population-averaged response for a wide range of stimulus, neuron, and synapse parameters. For recurrent systems, however, the exact shape of the rate transfer function at high frequencies can be crucial: Nordbø et al (2007), for example, have shown that the shape of the rate-transfer kernel determines the stability of persistent states in neural-field models. Solutions which are stable for exponential kernels can become unstable for alpha-function kernels.…”

“…(see Nordbø et al, 2007, and references therein) 6 . Note, that from this point of view the dynamical variable u ( t ) corresponds to the linear response of the firing rate.…”