Dynamical Systems in Neuroscience:

Simple Models 29920 mV100 ms49 pArecovery variablespiking limitcycleBABA-60 mVmembrane potentialFigure 8.16: Stuttering behavior of an RS neuron (data were provided by Dr. Klaus M.Stiefel, P28-36 adult mouse, coronal slices, 300µm, layer II/III pyramid, visual cortex).attraction domain of the focus (shaded region in the figure) is bounded by the stablemanifold of the saddle. As I increases, the stable manifold makes a loop and becomesa big homoclinic orbit giving birth to a spiking limit cycle attractor. When I = 125,stable resting and spiking states co-exist, which plays an important role in explainingthe paradoxical stuttering behavior of some neocortical neurons discussed later. AsI increases, the saddle quantity, i.e., the sum of eigenvalues of the saddle, becomespositive. When the stable manifold makes another, smaller loop, it gives birth toan unstable limit cycle, which then shrinks to the resting equilibrium and results insubcritical Andnronov-Hopf bifurcation.What is the excitability class of the RS model neuron in Fig. 8.15? If a slow ramp ofcurrent is injected, the resting state of the neuron becomes a stable focus and then losesstability via subcritical Andronov-Hopf bifurcation. Hence the neuron is a resonatorexhibiting Class 2 excitability. Now suppose steps of dc-current of amplitude I = 125pA or less are injected. The trajectory starts at the initial point (v, u) = (−60, 0),which is the resting state when I = 0, and then approaches the spiking limit cycle.Because the limit cycle was born via a homoclinic bifurcation to the saddle, it has alarge period and hence the neuron is Class 1 excitable. Thus, depending on the natureof stimulation, i.e., ramps vs. pulses, we can observe small or large spiking frequencies,at least in principle.In practice, it is quite difficult to catch homoclinic orbits to saddles because they aresensitive to noise. Injection of a constant current just below the neuron’s rheobase inFig. 8.15 would result in random transitions between the resting state and a periodicspiking state. Indeed, the two attractors co-exist and are near each other, so weakmembrane noise can push the trajectory in and out of the shaded region resulting ina stuttering spiking, illustrated in Fig. 8.16, mingled with subthreshold oscillations.Such a behavior is also exhibited by FS interneurons studied later in this section.