Neuronal Gamma Oscillations  Simulation

Design Stage 

1. Kickoff meeting with our mentor Dr. Kresimir Josic, Professor from the Department of Mathematics at the University of Houston.

2. Selected mathematical models: two-cell PING, RTM (excitatory neuron), WB (inhibitory neuron), and Wilson-Cowan (population of excitatory and inhibitory neurons)

3. The code was written in Python using Google Colab based on the reading of Introduction to Modeling Neuronal Dynamics book by Christoph Börgers.

   1. We first created our constants for both cells

   2. Then, we created the variables that are needed to create the communication between the two cells

   3. We created the functions

4. We modified the variables to validate the behavior of the neurons

5. We expanded it from a single synapse between 2 neurons to a population of a 1000 neurons

Analysis

Hodgkin and Huxley model

We modeled the potential, injected current, and synaptic current between an excitatory and an inhibitory neuron.

Wilson-Cowan model

We modeled the firing rate of the neuronal population (not individual ones). We worked with two populations of neurons: excitatory and inhibitory. Where E is the firing rate of excitatory neurons, and I is the firing rate of inhibitory neurons. The model proposes:

Being

and

Now that we have the firing rates of the populations, we can ask - what is the probability that one neuron fires? In this model is assumed that one neuron at time k fires with a probability:

where it depends if it is an excitatory or inhibitory neuron the case that we will consider

Results 

We simulated a gamma oscillation by connecting an excitatory and an inhibitory cell using the Hodgkin and Huxley model of individual neurons. Check it out here.

Wilson-Cowan Model of a population of a 1000 neurons.

Conclusions

• Lots of meetings!

• Hard to pin down a specific question (we learned so much in NMA!)

• We were able to implement models that initially seemed really daunting.

• We were able to successfully model single inhibitory and excitatory neurons and a population of 1000 neurons. Furthermore, we were able to study the network using different parameters values.