Dynamical Systems in Neuroscience:

Chapter 7Neuronal ExcitabilityNeurons are excitable in the sense that they are typically at rest but can fire spikesin response to certain forms of stimulation. What kind of stimulation is needed tofire a given neuron? What is the evoked firing pattern? These are the questionsconcerning the neuron’s computational properties, e.g., whether they are integrators orresonators, their firing frequency range, the spike latencies (delays), the co-existence ofresting and spiking states, etc. From the dynamical systems point of view, neurons areexcitable because they are near a bifurcation from rest to spiking activity. The type ofbifurcation, and not the ionic currents per se, determines the computational propertiesof neurons. In this chapter we continue our effort to understand the relationshipbetween bifurcations of the resting state and the neuro-computational properties ofexcitable systems.7.1 ExcitabilityA textbook definition of neuronal excitability is that a “subthreshold” synaptic inputevokes a small graded post-synaptic potential (PSP), while a “superthreshold” inputevokes a large all-or-none action potential, which is an order of magnitude larger thanthe amplitude of the subthreshold response. Unfortunately, we cannot adopt thisdefinition to define excitability of dynamical systems because many systems, includingsome neuronal models discussed in Chap. 4, have neither all-or-none action potentialsnor firing thresholds. Instead, we employ a purely geometrical definition.From the geometrical point of view, a dynamical system with a stable equilibrium isexcitable if there is a large-amplitude piece of trajectory that starts in a small neighborhoodof the equilibrium, leaves the neighborhood, and then returns to the equilibrium,as we illustrate in Fig. 7.1, left.In the context of neurons, the equilibrium corresponds to the resting state. Becauseit is stable, all trajectories starting in a sufficiently small region of the equilibrium, muchsmaller than the shaded neighborhood in the figure, converge back to the equilibrium.Such trajectories correspond to subthreshold PSPs. In contrast, the large trajectoryin the figure corresponds to firing a spike. Therefore, superthreshold PSPs are those225