C. Elegans Neural Network Model

This model is simulated in brian2, using some of the properties that accompany the C.elegans worm, such as the existence of electrical (gap junctions) and chemical connections, the total number of neurons and inhibitory/excitatory synapses.

## Introduction

The existence of power law in the degree distributions of the C.elegans network lead to our goal of discovering some sort of self-organizing criticality in the neuron network of the worm.

## Designing the model

The model is a leaky integrate-and-fire and it is comprised of 300 neurons. The ratio *E/I* (excitatory/inhibitory) is set to 3.4.
These are the differential equations used:

* d
v

d
t

=

g
e

g
i

−
(
v
−

E
l

)

t
a

u
m

{\displaystyle \frac{dv}{dt} = \frac{g\_e + g\_i - (v - E\_l)}{tau\_m} }
* d

g
e

d
t

=

−

g
e

t
a

u
e

{\displaystyle \frac{dg\_e}{dt} = \frac{-g\_e}{tau\_e} }
* d

g
i

d
t

=

−

g
i

t
a

u
i

{\displaystyle \frac{dg\_i}{dt} = \frac{-g\_i}{tau\_i} }

where

E

l

{\displaystyle E\_{l}}

 is the resting potential and

t
a

u
m

,
t
a

u
e

,
t
a

u
i

{\displaystyle tau\_m, tau\_e, tau\_i}

 are the time constants for the membrane and the excitatory and inhibitory connections respectively.
The neurons use an initial `push' that decays exponentially and is not considered part of the results.

## Simulation

We applied parameter exploration in order to find the best combination of variables that best describe our measures of Self-Organized Criticality. The model produced subcritical, critical or supercritical dynamics according to the connectivity parameters. Subcriticality resembles a random network and supercriticality is considered when the neurons spike simultaneously, creating very clear oscillations. The results are subcritical when the probability of an inhibitory synapse connection is bigger than the excitatory one. Similarly, supercriticality is observed when the probability of an excitatory connection is bigger than the probability of an inhibitory connection.

Subcriticality - the network resembles a random network.Criticality - the spiking neuron activity is constant and it creates avalanches.Supercriticality - all neurons spike and cool down at the same time.
## Results

In the case where the system appears to be critical, we measure the avalanche size and plot its distribution function. According to Hardstone, Richard, et al. and Poil, Simon-Shlomo et al. a Detrended Fluctuation Analysis(DFA) component between 0.5 and 1 and a power law exponent (

α

{\displaystyle \alpha}

) of 1.5 support the critical state.
For a small t, the simulation produced an

α

{\displaystyle \alpha}

 of 1.59 ± 0.037 and the CCDF resembles a scale-free distribution. However when simulating for a much larger value of t, the distribution is more likely to be exponential.

t=1000 ms,

α

{\displaystyle \alpha}

 = 1.59 ± 0.037 and FDA = 0.42t=10000 ms,

α

{\displaystyle \alpha}

 = 1.31 ± 0.001 and FDA = 0.30
The products of the model do not seem to agree with our hypothesis. One of the reasons why this happens is perhaps the simplicity of the model and the fact that many variables had to be semi-randomly assigned since we could find no documentation on them. In addition, a random network created with brian2, even when using specifically assigned probabilities, is not a scale-free network. The "rich get richer" concept of power law distribution cannot apply in C. Elegans, as the network has a finite size and a limited number of neurons to connect to.

## References

1. ↑ Brian 2 code repository. https://github.com/brian-team/brian2
2. ↑  Teixeira, Filipe Peliz Pinto, and Murray Shanahan. "Local and global criticality within oscillating networks of spiking neurons." 2015 International Joint Conference on Neural Networks (IJCNN). IEEE, 2015.
3. ↑ Hardstone, Richard, et al. "Detrended fluctuation analysis: a scale-free view on neuronal oscillations." Scale-free Dynamics and Critical Phenomena in Cortical Activity (2012): 75.
4. ↑ Poil, Simon-Shlomo, et al. "Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks." The Journal of Neuroscience 32.29 (2012): 9817-9823.
5. ↑ Alstott, Jeff, Ed Bullmore, and Dietmar Plenz. "powerlaw: a Python package for analysis of heavy-tailed distributions." PloS one 9.1 (2014): e85777.