Bursting Activity in Weakly Electric Fish - Computational Neuroscience: A Comprehensive Approach - page 232

burst, thus potentiating the somatic DAP, iii) a shorter refractory period for somatic
spikes compared to dendritic ones renders backpropagation conditional on the instan-
taneous firing rate, iv) the rate of the DAP potentiation, which is part of a positive
feedback loop in which dendritic spike broadening activates a persistent Na + cur-
rent, which further boosts depolarization. The slow dynamics of the persistent Na +
current largely control burst duration and burst frequency.

8.5.3 Reduced models of burst firing

Detailed biophysical models are powerful tools for probing the understanding of
cellular mechanisms at a microscopic scale. However, they are computationally too
complex for modeling of large networks or for analyzing the behavior of single cells
from a dynamical systems perspective. Having understood the key mechanisms, it is
often possible to reduce a detailed biophysical model to its essential components and
then apply dynamical systems analysis to the lower-dimensional model [101]. The
multi-compartmental model described above has undergone two such reductions,
first to a two-compartment model, termed a ghostburster for reasons explained in
more detail below [32], and then to an even simpler two-variable delay-differential-
equation model [71].
To model the generation of the somatic DAP, only a somatic and one dendritic
compartment representing the entire apical dendritic tree are needed ( Figure 8.11a)
[32]. Soma and dendrite were equipped with fast Na + channels, delayed rectifier K +
currents, and passive leak current. Current flow between the compartments followed
simple electrotonic gradients determined by the coupling coefficient between the two
compartments, scaled by the ratio of somatic to total model surface (see also [64,
80, 129]). Thus, the entire system was described by only six nonlinear differential
equations using modified Hodgkin/Huxley kinetics [54]. To achieve the relatively
longer refractory period of the dendrite [75], the time constant of dendritic Na +
inactivation was chosen to be longer than somatic Na + inactivation and somatic K +
activation. The key element for the burst mechanism was the introduction of a slow
inactivation variable for the dendritic delayed rectifier current, whose time constant
was set to about 5 times slower than the mechanisms of spike generation. In this
configuration, the two-compartment model reliably reproduced the potentiation of
the somatic DAP, which eventually triggers the firing of a spike doublet, the burst
termination due to failure of backpropagation, and the rapid onset of the AHP [32]
(see Figure 8.9b ).
To study the burst dynamics, the ghostburster model was treated as a fast-slow
burster [57, 101], separating it into a fast subsystem representing all variables related
to spike generation, and a slow subsystem representing the dendritic K + inactivation
variable, p d . The fast subsystem could then be investigated using the slow variable as
a bifurcation parameter. The dashed lines in Figure 8.11b show the quasi-static bifur-
cation diagram with maximum dendritic membrane voltage as a representative state
variable of the fast subsystem, and p d as the slow subsystem. For constant values of
p d >

p d 1 the fast subsystem
transitions to a period-two limit cycle. This corresponds to intermittent doublet firing

p d 1 , there exists a stable period-one solution. At p d =