A state-space characterization of reliability in FitzHugh-Nagumo neurons and networks via contraction analysis
A new contraction-theoretic state-space characterization of neuronal reliability is developed for FitzHugh-Nagumo models. It is shown that the state space of a neuron can be partitioned into two contraction regions separated by an expansion region where trajectory expansion may occur. The unforced resting equilibrium lies in a contraction region, and each full spike drives the state to the other contraction region and then back to the first. In-between, the state crosses the expansion region. Over any given interval, reliability is characterized as an average contraction property given by the balance of the contraction and expansion accumulated along the trajectories. An exact integral characterization of this balance and a simpler sufficient dwell-time condition are derived. The framework is then extended to networks of FitzHugh-Nagumo systems with an arbitrary affine voltage-coupling topology. A network of $n$ neurons is shown to possess $2^n$ contraction regions, separated by an expansion region, and network reliability is again characterized in terms of average contraction. A proof-of-concept excitatory-inhibitory network example is used to illustrate a connection between this characterization of reliability and regulation in neuronal networks. The network uses an internal model of a class of inputs to create a homeostatic regime where the network output is insensitive to the specific input within that class. Reliability is then seen as providing the asymptotic convergence properties necessary to make such a target regime attractive. Simulations of multi-stage cascade compositions of this elementary homeostatic motif illustrate how regulation can support oscillatory-signal propagation without phase-lag accumulation.