User:Jrm228/Term Paper Proposal
7 March 2017
In the paper, Simple Model of Spiking Neurons, by Eugene M. Izhikevich, he proposes that a mathematical model can be created that is both computationally efficient and biologically plausible to model neurons, which our group hopes to replicate. The Hodgkin-Huxley model of a neuron is a two-dimensional system based on the membrane potential of the neuron (v) and the membrane recovery variable (u), which provides feedback to “v” . This membrane recovery variable takes into consideration the activation of potassium currents and inactivation of sodium currents when a neuron is firing to return the neuron to an inactive state after spiking. By determining the values of four parameters that are dependent on the type of neuron and other circumstantial factors, the model can provide various replicas of models of individual neurons that have been determined experimentally. In addition, we will attempt to demonstrate that these different models can be expressed simultaneously, and in both series and parallel to each other for different types of spiking, to realistically simulate the behavior of real neurological tissue in the brain.
As previously mentioned, the model has two state variables and two differential equations. As the author did in his model, we will assume that the neuron firing will be primarily dependent on the membrane potential and membrane recover variable. The first equation is the change of the membrane potential of the neuron over time, and is dependent on the “synaptic or injected dc-current” as well as the membrane potential and the membrane recovery variable . The second, acting as the change in the recovery variable, is determined by the prior membrane potential and membrane recovery variable, and on two parameters. These parameters are the time scale of recovery, and the sensitivity of the recovery to the fluctuations in the membrane potential. They are both constants which are determined by the neuron and its surrounding environment. Finally, there is an after-spike resetting that occurs when the membrane potential is greater than 30 mV, which resets the value of the membrane potential based on a third constant parameter describing the new membrane potential created by potassium conductance. The recovery variable as changes based on the new sodium and potassium conductance, provided by the previous recovery variable and the final constant variable.
To replicate and expand this model, we must understand how each of the variables changes and remodel the differential equations. First, we will make the model described in the paper that describes a single neuron firing based on the two differential equations and four parameters. We will check that our model is working by recreating the graphs in Figures 1 and 2 from the paper, which show various types of spiking which have values provided by the text . Next, we shall be able to design a code which takes our model and allows for the easy manipulation of all the variables and parameters, most likely through use of the Manipulate function. Once we confirm that this code is working, we will move to create a system of 1000 coupled spiking neurons, as in Figure 3, to demonstrate that we can link each of our neurons together in a real-time simulation . Finally, we will alter our code to allow for changes in the types of neurons in patterns or randomly in the system, and attempt to model real systems based on experimental data on the patterns of different types of spikes within the brain.
- Izhikevich, Eugene. "Simple Model of Spiking Neurons". IEEE Transactions on Neural Networks. 14 (6): 1569–1572.