Dynamical Systems in Neuroscience:

Introduction 11may look like the one in Fig. 1.9a or b. Then it starts to fire tonic spikes, so its phaseportrait has a limit cycle attractor and it may look like the one in Fig. 1.9c, with whitecircle denoting an unstable resting equilibrium. Apparently, there is some intermediatelevel of injected current that corresponds to the transition from resting to sustainedspiking, i.e., from the phase portrait in Fig. 1.9b to Fig. 1.9c. What does the transitionlook like?From dynamical systems point of view, the transition corresponds to a bifurcationof neuron dynamics, i.e., a qualitative change of phase portrait of the system. Forexample, there is no bifurcation going from phase portrait in Fig. 1.9a to that inFig. 1.9b, since both have one globally stable equilibrium; the difference in behavior isquantitative but not qualitative. In contrast, there is a bifurcation going from Fig. 1.9bto Fig. 1.9c since the equilibrium is no longer stable and another attractor, limit cycle,appeared. The neuron is not excitable in Fig. 1.9a but it is in Fig. 1.9b simply becausethe former phase portrait is far away from the bifurcation and the latter is near.In general, neurons are excitable because they are near bifurcations from resting tospiking activity, so the type of the bifurcation determines the excitable properties of theneuron. Of course, the type depends on the neuron’s electrophysiology. An amazingobservation is that there could be millions of different electrophysiological mechanismsof excitability and spiking, but there are only 4, yes four, different types of bifurcationsof equilibrium that a system can undergo without any additional constraints, suchas symmetry. Thus, considering these four bifurcations in a general setup we canunderstand excitable properties of many models, even those that have not been inventedyet. What is even more amazing, we can understand excitable properties of neuronswhose currents are not measured and whose models are not known, provided thatwe can identify experimentally which of the four bifurcations the resting state of theneuron undergoes.The four bifurcations are summarized in Fig. 1.12, which plots the phase portraitbefore (left), at (center), and after (right) a particular bifurcation occurs. Mathematiciansrefer to these bifurcations as being of co-dimension-1 because we need to vary onlyone parameter, e.g., the magnitude of the injected dc-current I, to observe the bifurcationsreliably in simulations or experiments. There are many more co-dimension-2,3, etc., bifurcation, but they need special conditions to be observed. We discuss theselater in Chap. 6.Let us consider the four bifurcation and their phase portraits in the figure. Thehorizontal and vertical axes are the membrane potential with instantaneous activationvariable and a recovery variable, respectively. At this stage, the reader is not requiredto fully understand the intricacies of the phase portraits in the figure, since they willbe explained systematically in later chapters.• Saddle-node bifurcation. As the magnitude of the injected current or any otherbifurcation parameter changes, a stable equilibrium corresponding to the restingstate (black circle marked “node” in Fig. 1.12a) is approached by an unstableequilibrium (white circle marked “saddle”), they coalesce and annihilate eachother, as in Fig. 1.12a, middle. Since the resting state no longer exists, the