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R. P. Ignatius
time scale of k and sensitivity of k to w, respectively. The dimensionless parameters
c and d represent the after spike reset values of membrane potential w and membrane
recovery variable k, respectively.
To study the network dynamics, the directed neural network of C. elegans is used.
The adult C. elegans hermaphrodite has 302 neurons. Its neural network is organized
into two, a large somatic nervous system (282 neurons) and a small pharyngeal
nervous system (20 neurons). For the study presented here, the local network of 131
frontal neurons of the C. elegans is considered.
The network created is such that 99 neurons are made excitatory and the remaining
32 ones are made inhibitory interneurons. Excitatory neurons stimulate the neurotransmitters and increase the firing probability of other neurons connected to it (the
feedback is chosen to be positive). But the inhibitory neurons inhibit the neurotransmitters. Parameters of the neurons are chosen such that they vary randomly from
neuron to neuron.
The synaptic current to the ith neuron is described as follows:
I i = I dc + G syn
j
A j,i I f ired
(3)
where I dc is the external dc pulse to the neuron, ‘G syn ’ is the total conductance of the
synapse, j indicates the index of the fired neuron, A ji is the ( j, i)th entry of adjacency
matrix A of the network and I f ired represents the unit dc feedback current from the
jth fired neuron that is connected to ith neuron. I f ired assumes zero value in case
of no firing. Thus the second term in Eq. (3) represents the total feedback from the
network to the ith neuron. The inhibitory interneurons chosen are 7th to 13th, 27th
to 40th, 76th to 79th, 82th to 86th, 124th and 125th neurons of adjacency matrix of
C. elegans. Parameters of the neurons are chosen such that they vary randomly from
neuron to neuron.
As the dynamics is examined, the average firing rate of the network fluctuates and
that fluctuation increases with the increase in G syn . It is verified that the fluctuation
is present even when the parameters of all the neurons are same. But the amplitude of
fluctuations increases as randomness of parameters is incorporated. With the increase
in synaptic conductivity, the influence imparted on a particular neuron by other
neurons of the network increases and hence average firing rate of the neurons of the
network increases almost exponentially. This in turn increases the memory effect
(long-term potentiation or depression) produced by the feedback current, and hence,
fluctuations increase with the increase in strength of synaptic conductivity.
The parameters of individual excitatory neurons are selected such that they show
any one of the behaviours such as regular spiking, intrinsic bursting or chattering.
But when introduced in the network, all of them exhibit chaotic chattering behaviour
regardless of their parameter values. The dynamics of inhibitory neurons do not show
dependence on synaptic coupling, but the dynamics of excitatory neurons vary in
accordance with synaptic conductance. Even for slight variation in input current [15]
the neuron exhibits different spike patterns. Therefore, the neurons are very sensitive
to slight variations in input current. To confirm the chaotic nature of neurons in the
R. P. Ignatius
time scale of k and sensitivity of k to w, respectively. The dimensionless parameters
c and d represent the after spike reset values of membrane potential w and membrane
recovery variable k, respectively.
To study the network dynamics, the directed neural network of C. elegans is used.
The adult C. elegans hermaphrodite has 302 neurons. Its neural network is organized
into two, a large somatic nervous system (282 neurons) and a small pharyngeal
nervous system (20 neurons). For the study presented here, the local network of 131
frontal neurons of the C. elegans is considered.
The network created is such that 99 neurons are made excitatory and the remaining
32 ones are made inhibitory interneurons. Excitatory neurons stimulate the neurotransmitters and increase the firing probability of other neurons connected to it (the
feedback is chosen to be positive). But the inhibitory neurons inhibit the neurotransmitters. Parameters of the neurons are chosen such that they vary randomly from
neuron to neuron.
The synaptic current to the ith neuron is described as follows:
I i = I dc + G syn
j
A j,i I f ired
(3)
where I dc is the external dc pulse to the neuron, ‘G syn ’ is the total conductance of the
synapse, j indicates the index of the fired neuron, A ji is the ( j, i)th entry of adjacency
matrix A of the network and I f ired represents the unit dc feedback current from the
jth fired neuron that is connected to ith neuron. I f ired assumes zero value in case
of no firing. Thus the second term in Eq. (3) represents the total feedback from the
network to the ith neuron. The inhibitory interneurons chosen are 7th to 13th, 27th
to 40th, 76th to 79th, 82th to 86th, 124th and 125th neurons of adjacency matrix of
C. elegans. Parameters of the neurons are chosen such that they vary randomly from
neuron to neuron.
As the dynamics is examined, the average firing rate of the network fluctuates and
that fluctuation increases with the increase in G syn . It is verified that the fluctuation
is present even when the parameters of all the neurons are same. But the amplitude of
fluctuations increases as randomness of parameters is incorporated. With the increase
in synaptic conductivity, the influence imparted on a particular neuron by other
neurons of the network increases and hence average firing rate of the neurons of the
network increases almost exponentially. This in turn increases the memory effect
(long-term potentiation or depression) produced by the feedback current, and hence,
fluctuations increase with the increase in strength of synaptic conductivity.
The parameters of individual excitatory neurons are selected such that they show
any one of the behaviours such as regular spiking, intrinsic bursting or chattering.
But when introduced in the network, all of them exhibit chaotic chattering behaviour
regardless of their parameter values. The dynamics of inhibitory neurons do not show
dependence on synaptic coupling, but the dynamics of excitatory neurons vary in
accordance with synaptic conductance. Even for slight variation in input current [15]
the neuron exhibits different spike patterns. Therefore, the neurons are very sensitive
to slight variations in input current. To confirm the chaotic nature of neurons in the