Dynamical Systems in Neuroscience:

Electrophysiology of Neurons 45ear), semi-infinite, and satisfy Rall’s branching law (Rall 1959). Much of the insightcan be obtained via simulations, which typically substitute the continuous dendriticstructure in Fig. 2.19a by a network of discrete compartments in Fig. 2.19b. Dynamicsof each compartment is simulated by a Hodgkin-Huxley-type model, and the compartmentsare coupled via conductances. For example, if V s and V d denote the membranepotential at the soma and in the dendritic tree, as in Fig. 2.19c, thenC s ˙Vs = −I s (V s , t) + g s (V d − V s ) , and C d ˙Vd = −I d (V d , t) + g d (V s − V d ) ,where each I(V, t) represents the sum of all voltage-, Ca 2+ -, and time-dependent currentsin the compartment, and g s and g d are the coupling conductances that dependon the relative sizes of dendritic and somatic compartments. One can obtain manyspiking and bursting patterns by changing the conductances and keeping all the otherparameters fixed (Pinsky and Rinzel 1994, Mainen and Sejnowski 1996).Once we understand how to couple two compartments, we can do it for hundreds orthousands of compartments. GENESIS and NEURON simulation environments couldbe useful here, especially since they contain databases of dendritic trees reconstructedfrom real neurons.Interestingly, the somatic-dendritic pair in Fig. 2.19c is equivalent to a pair ofneurons in Fig. 2.19d coupled via gap-junctions. These are electrical contacts thatallow ions and small molecules to pass freely between the cells. Gap junctions areoften called electrical synapses, because they allow potentials to be conducted directlyfrom one neuron to another.Computational study of multi-compartment dendritic processing is outside of thescope of this book. We consider multi-compartment models of cortical pyramidal neuronsin Chap. 8 and gap-junction coupled neurons in Chap. 10.2.3.5 Summary of voltage-gated currentsThroughout this book we model kinetics of various voltage-sensitive currents using theHodgkin-Huxley gate modelI = ḡ m a h b (V − E)whereI (µA/cm 2 ) currentV (mV) membrane voltageE (mV) reverse potentialḡ (mS/cm 2 ) maximal conductancemprobability of activation gate to be openhprobability of inactivation gate to be openathe number of activation gates per channelbthe number of inactivation gates per channelThe gating variables m and n satisfy linear first order differential equations (2.9) and(2.10), respectively. We approximate the steady-state activation curve m ∞ (V ) by the