Neurons and Near-Death Spikes
137
2 Spatiotemporal Activities of a Pulse-Coupled Biological
Neural Network (PCNN)
We can study the spatiotemporal activities and effects of chaotic neurons on the
dynamics of pulse-coupled biological neural network (BNN) [12]. BNN used in the
study described here is that of the C. elegans. C. elegans is a soil worm. It is a
simple multicellular organism for which the complete neural network is identified.
Earlier studies reveal that human neuron networks are similar to that of C. elegans
in terms of network motifs, types of synapses and neurons, etc. So the dynamics of
C. elegans neuronal network can be translated to that of mammals [13]. Just like
humans the nervous system of C. elegans contains both chemical synapses and gap
junctions. Also, neurons in them are arranged as modules. Genome of C. elegans is
35% similar to that of humans. These properties make them prototypes for studying
simple dynamics of human brain.
One-dimensional model of neural network can be used to study the dynamics of
entire network of C. elegans. On application of sinusoidal external input to neurons,
the dynamics for different values of external input and synaptic connections can be
studied and it has been identified that the network exhibits fixed point, periodic,
quasi-periodic and chaotic dynamics. The study presented here concentrates on spatiotemporal dynamics of the network and on the influence of chaotic neurons on
it.
2.1 Izhikevich Model of Neuron
Izhikevich neuron is a simple model, capable of displaying wide range properties of
real biological neurons. Computational minimalism of this model makes it suitable
for large-scale network simulation. The neuron model is represented by the following
equations [14].
w
= 0.04w
2
+ 5w + 140 − k + I
(1)
k
= a(bw − k)
with the auxiliary after-spike resetting given as
if w ≥ 30mV, w ← c and k ← k + d
(2)
where ‘w’ represents the membrane potential, k accounts for the activation of K+
ionic currents and inactivation of the Na
+ ionic currents. When membrane potential
reaches the maximum value, 30 mV, the membrane potential and current are reset
according to Eq. (2). The input current (synaptic current) to the neuron is delivered
through the variable I . In Eq. (1), dimensionless parameters a and b represent the
137
2 Spatiotemporal Activities of a Pulse-Coupled Biological
Neural Network (PCNN)
We can study the spatiotemporal activities and effects of chaotic neurons on the
dynamics of pulse-coupled biological neural network (BNN) [12]. BNN used in the
study described here is that of the C. elegans. C. elegans is a soil worm. It is a
simple multicellular organism for which the complete neural network is identified.
Earlier studies reveal that human neuron networks are similar to that of C. elegans
in terms of network motifs, types of synapses and neurons, etc. So the dynamics of
C. elegans neuronal network can be translated to that of mammals [13]. Just like
humans the nervous system of C. elegans contains both chemical synapses and gap
junctions. Also, neurons in them are arranged as modules. Genome of C. elegans is
35% similar to that of humans. These properties make them prototypes for studying
simple dynamics of human brain.
One-dimensional model of neural network can be used to study the dynamics of
entire network of C. elegans. On application of sinusoidal external input to neurons,
the dynamics for different values of external input and synaptic connections can be
studied and it has been identified that the network exhibits fixed point, periodic,
quasi-periodic and chaotic dynamics. The study presented here concentrates on spatiotemporal dynamics of the network and on the influence of chaotic neurons on
it.
2.1 Izhikevich Model of Neuron
Izhikevich neuron is a simple model, capable of displaying wide range properties of
real biological neurons. Computational minimalism of this model makes it suitable
for large-scale network simulation. The neuron model is represented by the following
equations [14].
w
= 0.04w
2
+ 5w + 140 − k + I
(1)
k
= a(bw − k)
with the auxiliary after-spike resetting given as
if w ≥ 30mV, w ← c and k ← k + d
(2)
where ‘w’ represents the membrane potential, k accounts for the activation of K+
ionic currents and inactivation of the Na
+ ionic currents. When membrane potential
reaches the maximum value, 30 mV, the membrane potential and current are reset
according to Eq. (2). The input current (synaptic current) to the neuron is delivered
through the variable I . In Eq. (1), dimensionless parameters a and b represent the