100 ms

Figure 8 Thalamic relay neurons show two distinct firing modes. Shown are simulated responses of a thalamic relay neuron [Figure generated using the computational model from D. A. McCormick and J. R. Huguenard (1992). A model of the electrophysiological properties of thalamocortical relay neurons. J. Neurophysiol. 68, 1384-1400]. (a) Under hyperpolarized conditions, relay neurons fire in an oscillatory manner. Slow oscillations are generated by interactions of the transient Ca2 + current It and the slow, hyperpolarization-activated cation current Ih. Fast action potentials, mediated by Na+ and K+, occur at the peaks of the Ca2+ spikes. (b) Depolarization by any of a number of means (e.g., current injection or neuromodulation) puts the neuron in a "tonic firing'' mode. In this mode, It is inactivated and Ih is deactivated. Consequently, the cell's behavior is dominated by the Na+ and K+ currents exclusively.

based on the techniques of nonlinear dynamics have been successful in identifying the particular features that determine a host of important features of neuronal excitability, including threshold, the relationship between sustained firing rates and applied current, and particular patterns of bursting.

Figure 9 shows how nonlinear dynamics can be applied to understand neuronal threshold. Figures 9a and 9b show results from simulations of the Hodgkin-Huxley equations with zero applied current but two different initial conditions in membrane potential. As Figs. 9a and 9b demonstrate, excitable cells can be exquisitely sensitive to initial conditions. Examining the time derivative of membrane potential Vm(t) shortly after the perturbation in initial conditions gives insight into this phenomenon. Figure 9c shows results from hundreds of simulations conducted over a large range of initial values V0. Vm(t), evaluated at t = 0.5 msec, is plotted vs V0 (solid line; the value t = 0.5 msec was chosen because this is long enough for the m gate to react significantly to the perturbation away from resting potential but short enough that the h or n gates remain relatively near their resting values). Also plotted are the contributions to Vm(0.5) from Na+ conductance (dashed line) as well as K+ and leak conductances (dotted line). These contributions can be obtained easily from the Hodgkin-Huxley equation describing Vm(t). The effect of the Na+ conductance is to elevate Vm(t); the effect of the K+ conductance is to reduce Vm(t).

Figure 9d shows a magnified view of Vm(0.5) in the region near spike threshold. The plot shows two zero-crossings, at V0 = —65 and — 59 mV. These zero crossings, often called fixed points, are especially important because where Vm(t) = 0, membrane potential Vm is by definition not changing, meaning that Vm has the potential to remain fixed at that point indefinitely (for a system with constant parameters). The slope of the curve at each zero crossing tells us much about the stability of that fixed point in response to small fluctuations (e.g., due to noise). For example, consider the special case of no perturbation. In this case, V0 equals resting potential (—65 mV), which we expect to be a fixed point. For Vm slightly less than —65 mV, Vm(0.5)>0, implying that Vm will return to its resting value. For Vm slightly higher than —65 mV, Vm(0.5)<0, implying again that Vm will return to a. V = -58 mV

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