# Model Neurons and Fast-Slow Systems

### Introduction

Neurons are the fundamental units of information processing in the human body and as such have been the subject of much study. A. L. Hodgkin and A. F. Huxley in the early 1950's found that a neuron processes information by controlling the flow of charged ions through its cell membrane, hence generating an electrical signal. Hodgkin and Huxley proposed that a neuron can be modeled via an equivalent electrical circuit and ultimately by a system of four differential equations. This work is the foundation of modern neuroscience and earned them the Nobel Prize. Since this pioneering work more and more detailed models of specific types of neurons have been developed through a combination of experiments and mathematics. The advent of the digital computer has allowed the numerical study of these models, providing new insights into the function of neurons.

A neuron transmits information by generating action potentials.'' In general, neurons exhibit two modes of operation, excitable and bursting. An excitable neuron is quiet (does not generate action potentials) except in response to an outside stimulus (for example from a second neuron through a synapse). A bursting neuron generates periodic trains of action potentials. This is often the case in neurons that help regulate rhythmic body functions such as heart contractions. Many neurons exhibit both of these behaviors depending on various factors. A common experiment is to apply different levels of current through an electrode to a neuron to determine whether and where a neuron transitions from excitable to bursting.

### The Model

In this lab we will study the Morris-Lecar equations, a system of two differential equations similar to the Hodgkin-Huxley equations, to explore how a neuron responds to external stimulus. The Morris-Lecar equations were originally formulated to describe electrical activity in barnacle muscle fiber and are sometimes used as a simple caricature of the envelope of bursting neurons and only explicitly model the flow of potassium (K+) and calcium (Ca2+) ions. The variable v in the given equations denotes the voltage of the neuron while the variable w is known as a recovery variable and describes the percentage of open channels selectively permeable to Ca2+.

v' = I + 2w(-0.7-v) + 0.5(-0.5-v) + 1.1m(v)(1-v)
w' = ελ(v)(w(v) - w)

The functions m(v), w(v), and λ(v) are given by

The parameter I represents injected current into the model neuron. Both I and ε will be treated a parameters in the exercises. The values of the other constants are given in the table below.

Constant

Value

-0.01

0.15

-0.12

0.30

0.22

2.00

-0.70

0.50

-0.50

1.10

Problem 1. Plot the functions m(v) and w(v). What happens to the graphs of these functions as v2 and v4 approach zero respectively?

Problem 2. Compute and .

Problem 3. Speculate on the purpose of these functions in these equations.

### Analysis

Excitable Neurons.
Enter the Lecar-Morris equations into your ordinary differential equations solver. For your convenience, a phase plane is displayed below for the equations, if you prefer to use it instead of your own solver.

Begin by setting I = 0.25 and ε = 0.1. Your phase space window should be -1 ≤ v ≤ 1, -0.5 ≤ w ≤ 1. Set your integrator to integrate for a total time of t=40. You might want to use an integrator specifically designed for stiff'' differential equations such as Gear's method.

Problem 1. Draw both the v and w nullclines and compute the stability of the critical point.

Problem 2. Compute the solutions to the Morris-Lecar equations with initial conditions of (-0.5,0.1) and (-0.5,0.2). Plot these solutions both in the (v,w) phase plane and plot the graphs of v vs t. For each initial condition describe the relationship between the trajectory in phase space and the v vs t plot. What are the differences and similarities between the two solutions? What is the physical interpretation?

Problem 3. Keeping the graphs of the nullclines on the screen draw trajectories from at least 6 initial conditions on a vertical line left of the critical point extending beyond the minimum of the v-nullcline. Describe what you see. What do you think the physical interpretation of this phenomenon is?

Problem 4. In the previous exercise you noticed that some points on your line went directly to the critical point while others made a long excursion around the right branch of the v nullcline before approaching the critical point. This phenomenon extends through much of the phase plane. Try to determine a curve or region that partitions the phase plane by this criterion. This may have to be done by a combination of computing and using a printout of your nullclines to record how each initial condition behaves. Are there any points that don't fit into either category?

Problem 5. Repeat exercise (3) for ε = 0.05 and ε = 0.02 (you may need to increase your total integration time as ε decreases). How does the separation between the two regions change? What do you think separates theses regions in the limiting case ε → 0.

You probably noticed that for small ε trajectories consisted of fast, almost horizontal jumps'' to the v-nullcline and slow drifts'' along the v-nullcline. This behavior is typical of differential equations of the form

 x' = f(x,y) (1) y' = ε g(x,y)

for 0 < ε << 1. Note that if ε ≡ 0 then y' ≡ 0 and hence y acts as a parameter.

Problem 1. What is the interpretation of the v-nullcline when ε ≡ 0 in the Morris-Lecar equations? Describe the dynamics for each value of w. What can you say about these dynamics and the dynamics you observed numerically for small ε?

Problem 2. Let τ = εt and use the chain rule on equations (1) to show that

If ε ≡ 0 this becomes

If we can solve (2) for x in terms of y then there exists a function such that . In other words, the x-coordinate of the flow is given by . This means that the flow is on the curve with velocity given by . The curve is known as the slow manifold and the flow on this manifold is known as the slow flow. For the Morris-Lecar equations the Implicit Function Theorem can be used to show that there exists functions and that parameterize the left and right branches of the v-nullcline in terms of w. You won't be able to compute these functions, however the qualitative dynamics on each of these branches can be determined by considering the geometry of both nullclines. Do this to show that the slow flow on the left branch is towards the critical point at the intersection of the two nullclines and that the flow on the right branch is upward to the local maxima of the v-nullcline.

Problem 3. The singular dynamics'' are determined by combining the results of parts one and two. Describe the singular orbit originating from the points (-0.5,0.1) and (-0.5,0.2). How do these compare with the numerically computed orbits originating from the same points for small ε? Numerically compute the orbits from these points for various values of ε. At what value of ε do you think the singular approximation ceases to be valid?

Mathematical modeling of Biological Neural Networks

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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.