Dynamical Systems in Neuroscience:

250 Excitabilitysquid axonmodel0 mV0 mV-50 mV-50 mV5 ms5 msFigure 7.26: Variable-size action potentials in squid giant axon and revised Hodgkin-Huxley model (Clay 1998) in response to brief steps of currents of variable magnitudes(data was kindly provided by John Clay).Conversely, if the neuron was briefly hyperpolarized and then released from hyperpolarization,it could fire a rebound post-inhibitory spike, as we discuss later in thischapter (see Fig. 7.29). Apparently, releasing from hyperpolarization does not qualifyas a superthreshold stimulation. Why then the neuron fired?7.2.5 Threshold manifoldsThe problem of formulating a mathematical definition of firing thresholds was firsttackled by FitzHugh (1955). Using geometrical analysis of neural models, he noticedthat thresholds, if exist, are never numbers but manifolds, e.g., curves in twodimensionalsystems. We illustrate his concept in Fig. 7.27 using phase plane analysisof the I Na,p +I K -model.Integrators do have well-defined threshold manifolds. Since an integrator neuron isnear a saddle-node bifurcation, whether on or off an invariant circle, there is a saddlepoint with its stable manifold; see Fig. 7.27a. This manifold separates two regions ofthe phase space, and for this reason often called a separatrix. Depending on the prioractivity of the neuron and the size of the input, its state can end up in the shadedarea and generate a subthreshold potential, or in the white area and generate an actionpotential. An intermediate-size input cannot reduce the size of the action potential,but can only delay its occurrence. In the extreme case, a perturbation can put the statevector precisely on the threshold manifold, and the system converges to the saddle, atleast in theory. Since the saddle is unstable, small noise present in neurons pushes thestate either to the left or to the right, resulting in either a long subthreshold potentialor a large-amplitude spike with a long latency, as we discuss in Sect. 7.2.9 and showin Fig. 7.34. Finally, notice that a neuron has a single threshold value of membranepotential only when its threshold manifold is a straight vertical line.Resonators may or may not have well-defined threshold manifolds, depending onthe type of bifurcation. Consider a resonator neuron in the bistable regime; that is,sufficiently near a subcritical Andronov-Hopf bifurcation with an unstable limit cycle