Multistability in neural networks with delayed feedback: Theory and applications

In this dissertation, we study the coexistence of multiple stable patterns (multistability) in recurrent neural networks with delayed feedback. The coexistence of multiple stable patterns such as equilibria and periodic orbits is the basis for associative content-addressable memory storage and retrieval in neural networks where each equilibrium is identified with static memory, while stable periodic orbits are associated with temporally patterned spike trains. Periodic patterns exhibited in recurrent neural networks are also associated with a variety of rhythms displayed in the nervous system. These rhythms have been linked to important behavioral and cognitive states in the nervous system, including attention, working memory, associative memory, object recognition, sensory motor integration and perception processing.;Our investigation addresses the questions of how we can efficiently increase the network's capacity for memory storage and retrieval, and what kind of mechanisms enable neural networks to generate a large number of coexisting stable oscillatory patterns. These questions are addressed in this dissertation from two aspects: (1) the incorporation of some important biological features of neurons such as the firing process and the absolute refractory period, (2) the introduction of a non-monotonic activation function. Our results show that the interaction of time lag, the recurrent feedback, biological features of individual neurons, and the non-monotonicity of synaptic update functions leads to a large number of stable periodic solutions with predictable patterns of oscillations, via interesting pattern transition or through a mode interaction of pitchfork, saddle-node and Hopf bifurcations.;First, we investigate the impact of the effective duration of a delayed feedback on multistability in a recurrent inhibitory loop when biological realities of firing and absolute refractory period are incorporated into an integrate-and-fire neuron model. Our analysis shows that the interaction of the delay, the inhibitory feedback and the absolute refractory period can generate four basic types of oscillations which give the basic building blocks of possible periodic patterns. We then show how these basic oscillations can be pinned together to form four types of periodic patterns, such as self-inhibitory patterns and nearest-neighbor-inhibitory patterns. The coexistence of these different types of periodic patterns leads to the occurrence of multistability in the recurrent inhibitory loop. Moreover, interesting pattern transitions occur as the time delay passes through certain critical values. These pattern transitions play a similar role to the standard bifurcation theory in terms of the birth and continuation of multiple periodic patterns. We also link the identified periodic patterns to certain neural rhythms, and use the average time of convergence to a periodic pattern to determine what kind of periodic patterns have the potential to be used for neural information transmission and cognition processing in the nervous system.;Second, we examine the effect of non-monotonic activation functions on the network's capacity for memory storage and retrieval. We first show how supercritical pitch-fork bifurcations and saddle-node bifurcations lead to the coexistence of multiple stable equilibria in the instantaneous updating network. We then study the effect of time delay on the local stability of these equilibria and show that four equilibria lose their stability at a certain critical value of time delay, and a Hopf bifurcation of four periodic solutions occurs, leading to multiple coexisting periodic orbits. We apply center manifold theory and normal form theory to determine the direction of this Hopf bifurcation and the stability of bifurcated periodic orbits. Numerical simulations show very interesting global patterns of periodic solutions as the time delay is varied. In particular, we observe that these four periodic solutions are glued together along the stable and unstable manifolds of saddle points to develop a butterfly structure through a complicated process of gluing bifurcations of periodic solutions with increasing frequencies crossing the stable manifolds of the saddle points.