Transient Dynamics in Neural Networks

The motivation for this thesis is to devise a simple model of transient dynamics in neural networks. Neural circuits are capable of performing many computations without reaching an equilibrium, but instead through transient changes in activity. Thus, having a good model for transient activity is important. In particular, this thesis focuses on a firing-rate description of neural activity. Firing rates offer a convenient simplification of neural activity, and have been shown experimentally to convey information about stimuli and behavior. This work begins by review the philosophy of modeling firing rates, as well as the problems that go with it. It examines traditional approaches to modeling firing rates, and in particular how common assumptions lead to a model that fails to capture transient dynamics. Chapter 2 applies a traditional model of firing rates in order to gain insight into properties of cortical circuitry. In collaboration with the lab of David Ferster at Northwestern University, we found that surround suppression in cat primary visual cortex is mediated by a withdrawal of excitation in the cortical circuit. In theoretical work, we find that this behavior can only arise if excitatory recurrence alone is strong enough to destabilize visual responses but feedback inhibition maintains stability. Chapter 3 reviews concepts and literature related to the dynamics of large networks of spiking neurons. Population density approaches are common for describing the dynamics of networks of spiking neurons. These approaches allow for a rigorous approach to relate the dynamics of individual neurons to the population firing rate. Chapter 4 explores a method for accurately approximating the firing-rate dynamics of a population of spiking neurons. We describe the population by the probability density of membrane potentials, so the dynamics are governed by a Fokker-Planck equation. Using a spiking model with periodic boundary conditions, we write the Fokker-Planck dynamics in a Fourier basis. We find that the lowest Fourier modes dominate the dynamics. Chapter 5 presents a novel rate model that successfully captures synchronous dynamics. As in the previous chapter, we invoke an approximation to the dynamics of a population of spiking neurons in order to develop a firing-rate model. Our approach derives from an eigenfunction expansion of a Fokker-Planck equation, which is a common approach to solving such problems. We find that a very simple approximation turns out to be surprisingly accurate. This approximation allows us to write a closed-form expression for the firing rate that resembles the equations for a damped harmonic oscillator. Finally, chapter 6 uses the formalism derived in the previous chapter to analyze activity in a large randomly-connected network of neurons. Comparing this large spiking network to a network of two coupled rate units, we find that the firing rate network gives a good approximation to the time-varying activity of a spiking network across a wide range of parameters. Perhaps most surprisingly, we also find that the rate network can approximate the phase diagram of the spiking network, predicting the bifurcation line between synchronous and asynchronous states.