Oscillatory Dynamics in Biological Neurons - Advanced Models of Neural Networks - page 76

current which is activated or deactivated at a slower pace than the rest of the currents.
A slow current which forms the basis for the slow modulation phase is the calcium-
dependent potassium current I KCa . Ions of Ca 2C enter the cell's membrane during
the active spiking phase and this results in raise (activation) of the current I KCa
(outward current). When the outward current becomes sufficiently large the cell can
no longer produce the spiking activity and the active phase is terminated. During the
silent phase, there is an outflow of Ca 2C from the cell's membrane and the calcium
channels close.
A generic mathematical model of the bursting activity is described by a set of
differential equations which evolve at multiple scales

dx
dt

D f.x;y/

(4.1)

dy
dt

D g.x;y/

where >0takes a small positive value. In the above set of equations x2R n are fast
variables which are responsible for spikes generation, and y2R m are slow variables
which are responsible for the slow modulation of the silent and active phases.
There are different types of bursting which are different in the amplitude and
frequency of oscillations that the neuron's membrane performs during the active
phase. These are [ 3 , 16 , 49 , 65 ]:

1. square-wave bursting: This form of bursting was first observed in pancreatic ˇ-
cells. In square-wave bursting the amplitude of the membrane's voltage variation
remains practically unchanged, whereas the frequency of spiking slows down
during the active phase.
2. elliptic bursting: In elliptic bursting one can notice small amplitude oscillations
during the silent phase, and progressively the amplitude of the oscillations
increases and remains unchanged for a long time interval. During the active
phase the frequency of spikes initially increases and subsequently drops. Elliptic
bursting appears in models of thalamic neurons, and particularly in neurons of
the basal ganglia.
3. parabolic bursting: In parabolic bursting there are no oscillations during the silent
phase and there is an abrupt transition to the active phase which consists of
oscillations of large amplitude. The values of the parameters of the dynamic
model of the neuron are the ones which define which type of bursting will finally
appear.

Indicative diagrams of biological neurons under spiking activity are given in
Fig. 4.1 . Moreover, representative diagrams of neurons under bursting activity are
shown in Fig. 4.2 .
It is noted that by changing the parameters of the dynamic model, the neuron
bursting can exhibit chaotic characteristics which have to do with the number of
spikes during the active phase, the frequency under which bursts appear and the
transition from bursting to spiking [ 88 ].