Dynamical Systems in Neuroscience:

Two-Dimensional Systems 99108642y0-2-4-6-8-10-10 -8 -6 -4 -2 0 2 4 6 8 10xFigure 4.6: Representative trajectories of the two-dimensional system (4.3,4.4).going. Thus, to understand the geometry of a vector field, it is always useful to plot afew representative trajectories starting from various initial points, as we do in Fig. 4.6.Due to the uniqueness of the solutions, the trajectories cannot cross, so they partitionthe phase space into various regions. This is an important step toward determiningthe phase portrait of a two-dimensional system.Let us return to the I Na,p +I K -model (4.1, 4.2) with low-threshold K + current andexplain two odd phenomena discussed in the first chapter: Failure to generate all-ornoneaction potentials (Fig. 1.5b) and inability to have a fixed value of the thresholdvoltage. Brief and strong current pulses in Fig. 4.7 reset the value of the voltage variableV but do not change the value of the K + activation variable n. Thus, each voltagetrace after the pulse corresponds to a trajectory starting with different values of V 0 butthe same value n 0 . We see that each trajectory makes a counter-clockwise excursionand returns to the rest state. However, the size of the excursion depends on the initialvalue of the voltage variable and can be small (subthreshold response), intermediate, orlarge (action potential). This phenomenon was considered theoretically by FitzHugh inearly sixties (see bibliography) and demonstrated experimentally by Cole et al. (1970)using squid giant axon at higher than normal temperatures.In Fig. 4.8 we apply a long pre-pulse current of various amplitudes to reset theK + activation variable n to various values, and then a brief strong pulse to reset Vto exactly −48 mV. Each voltage trace after the pulse corresponds to a trajectorystarting with the same V 0 = −48 mV, but different values of n 0 . We see that sometrajectories return immediately to the rest state while others do so after generating anaction potential. Therefore, V = −48 mV is a subthreshold value when n 0 is large,and a superthreshold one otherwise.