Abstract
We investigate the dynamics of recurrent networks of excitatory (E) and inhibitory (I) neurons in the presence of time-dependent inputs. The dynamics is characterized by the network dynamical transfer function, i.e., how the population firing rate is modulated by sinusoidal inputs at arbitrary frequencies. Two types of networks are studied and compared: (i) a Wilson–Cowan type firing rate model; and (ii) a fully connected network of leaky integrate-and-fire (LIF) neurons, in a strong noise regime. We first characterize the region of stability of the “asynchronous state” (a state in which population activity is constant in time when external inputs are constant) in the space of parameters characterizing the connectivity of the network. We then systematically characterize the qualitative behaviors of the dynamical transfer function, as a function of the connectivity. We find that the transfer function can be either low-pass, or with a single or double resonance, depending on the connection strengths and synaptic time constants. Resonances appear when the system is close to Hopf bifurcations, that can be induced by two separate mechanisms: the I–I connectivity and the E–I connectivity. Double resonances can appear when excitatory delays are larger than inhibitory delays, due to the fact that two distinct instabilities exist with a finite gap between the corresponding frequencies. In networks of LIF neurons, changes in external inputs and external noise are shown to be able to change qualitatively the network transfer function. Firing rate models are shown to exhibit the same diversity of transfer functions as the LIF network, provided delays are present. They can also exhibit input-dependent changes of the transfer function, provided a suitable static non-linearity is incorporated.
Highlights
Networks of neurons in the central nervous system are driven by stimuli that vary on a wide range of time scales, and need to encode these stimuli by the pattern of firing of their constituent neurons
3 Analytical Methods The analysis of the network response is performed in three steps. (1) we compute the background activity as a function of the mean drive; (2) we check whether background activity with a constant firing rate is stable; (3) we compute the dynamical response of the network
The stationary firing rate r0 is given by r0
Summary
Networks of neurons in the central nervous system are driven by stimuli that vary on a wide range of time scales, and need to encode these stimuli by the pattern of firing of their constituent neurons To understand how this encoding is performed, one needs to understand the relationship between the input to the network (the set of spike trains of all neurons that are pre-synaptic to a given network) and its output (the set of spike trains of all neurons in the network). A simpler question, and a useful first step, is to understand the relationship between the mean inputs to a network and its population firing rate (i.e., average instantaneous probability that a neuron in the network emits an action potential) Even this question can be difficult to answer in the space of all possible time-dependent inputs. These two quantities together characterize the transfer function of the network, and can be used to reconstruct the dynamics of the population in response to arbitrary time-dependent inputs, provided the amplitude of the time-dependent variations in the input is sufficiently small
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