Bursting Activity in Weakly Electric Fish - Computational Neuroscience: A Comprehensive Approach - page 234

Figure 8.11
Two-compartment model of burst generation. a) Sketch of the somatic and dendritic
compartments linked by an axial resistance. b) The dashed lines show the quasi-
static bifurcation diagram with a representative of the fast subsystem, the maximum
dendritic membrane voltage, as a function of the slow subsystem, the dendritic K +
inactivation variable, p d . Overlaid is a single burst trajectory (solid line; burst begins
with the upwards pointing arrow on the right). Adapted from [32].

with dendritic spike failure, since for p d < p d 1 dendritic repolarization is sufficiently

slow to cause very strong somatic DAPs capable of eliciting a second somatic spike
after a small time interval ( 3 ms). The overlaid burst trajectory (solid line) shows

the beginning of the burst on the right side (upwards arrow). The maximum of the
dendritic membrane voltage decreases for the second spike of the doublet (compare
Figure 8.9b ), which occurs at p d <

∼

p d 1 . The short doublet ISI is followed by the long
interburst ISI, the slow variable recovers until the next burst begins. Because p d is
reinjected near an infinite-period bifurcation (saddle-node bifurcation of fixed points
responsible for spike excitability), Doiron et al. [32] termed this burst mechanism
ghostbursting ( sensing the ghost of an infinite-period bifurcation [118]). Thus, the
two-compartment model nicely explains the dynamics of pyramidal cell bursting ob-
served in vitro by the interplay between fast spike-generating mechanisms and slow
dendritic K + -channel inactivation.
In a further reduction of the model, Laing and Longtin [71] replaced the six ordi-
nary differential equation model by an integrate-and-fire model consisting of a set of
two discontinuous delay-differential equations. An interesting aspect of this model
is that it uses a discrete delay to mimic the ping-pong effect between soma and den-