III Networks of Neurons and Population Activity

Chapter 13 Continuity Equation and the Fokker-Planck Approach

In the previous Chapter, the notion of a homogeneous population of neurons has been introduced. Neurons within the population can be independent, fully connected, or randomly connected, but they all should have identical, or at least similar, parameters and all neurons should receive the same input. For such a homogeneous population of neurons, it is possible to predict the population activity in the stationary state of asynchronous firing (Section 12.4 in Ch. 12). While the arguments we made in the previous chapter are general and do not rely on any specific neuron model, they are unfortunately restricted to the stationary state.

In a realistic situation, neurons in the brain receive time-dependent input. Humans change their direction of gaze spontaneously two or three times per second. After each gaze change, a new image impinges on the retina and is transmitted to visual cortex. Auditory stimuli such as music or traffic noise have a rich intrinsic temporal structure. If humans explore the texture of a surface which by itself is static, they move their fingers so as to actively create temporal structure in the touch perception. If we think back of our last holiday, we recall sequences of events rather than static memory items. When we type a message on a keyboard, we move our fingers in a rapid pattern. In none of these situations, stationary brain activity is a likely candidate to represent our thoughts and perceptions. Indeed, EEG (electroencephalography) recordings from the surface of the human scalp, as well as multi-unit activity recorded from the cortex of animals, indicate that the activity of the brain exhibits a rich temporal structure.

In this chapter, we present a formulation of population activity equations that can account for the temporal aspects of population dynamics. It is based on the notion of membrane potential densities for which a continuity equation is derived (Section 13.1). In order to illustrate the approach, we consider a population of neurons receiving stochastically arriving spikes (Sections 13.2 and 13.3). For an explicit solution of the equations, we first focus on coupled populations of leaky integrate-and-fire neurons (Sections 13.4), but the mathematical approach can be generalized to arbitrary non-linear integrate-and-fire neurons (Section 13.5) and generalized integrate-and-fire neurons with adaptation (Section 13.6).

Before we turn to neurons with adaptation, we focus on one-dimensional, but potentially nonlinear integrate-and-fire neurons. Knowledge of the momentary membrane potential and the input is sufficient to predict the future evolution of a single integrate-and-fire neuron. In a large population of neurons, the momentary state of the population as a whole can therefore be characterized by the momentary distribution of membrane potentials. The evolution of this distribution over time is summarized by the continuity equation which is introduced now.