Dynamical Systems in Neuroscience:

Two-Dimensional Systems 103derivative, V' (mV/ms)membranepotential (mV)2001000-100-80 -60 -40 -20 0 20 40membrane potential, V (mV)40200-20-40cortical pyramidal neuron cortical interneuron brainstem neuron-600 50 100-80 -60 -40 -20 0 20 40membrane potential, V (mV)-80 -60 -40 -20 0 20 40membrane potential, V (mV)0 20 40 60 0 10 20time (ms)Figure 4.11: Limit cycles corresponding to tonic spiking of three types of neuronsrecorded in vitro.ẏ = µg(x, y) (slow variable)where small parameter µ describes the ratio of time scales of variables x and y. Typically,fast variable x has a cubic-like nullcline that intersects the y-nullcline somewherein the middle branch, as in Fig. 4.12a, resulting in relaxation oscillations. The periodictrajectory of the system slides down along the left (stable) branch of the cubic nullclineuntil it reaches the left knee A. At this moment, it quickly jumps to the point B andthen slowly slides up along the right (also stable) branch of the cubic nullcline. Uponreaching the right knee C, the system jumps to the left branch and starts to slide downagain, thereby completing one oscillation. Relaxation oscillations are easy to graspconceptually, but some of their features are quite difficult to study mathematically.We consider relaxation oscillations in detail in Sect. 6.3.4.Notice that the jumps in Fig. 4.12a are nearly horizontal — a distinctive signature ofrelaxation oscillations that is due to the disparately different time scales in the system.Although many neuronal models have fast and slow time scales and could be reducedto the fast/slow form above, they do not exhibit relaxation oscillations because theparameter µ is not small enough. Anybody who records from neurons would probablynotice the weird square shape of “spikes” in Fig. 4.12b, something that most biologicalneurons do not exhibit. Nevertheless, relaxation oscillations in fast/slow systems areimportant when we consider neuronal bursting in Chap. 9, though the fast variable xis two-dimensional there.