Preliminary Information

General Background

Nearly all of single neuron modeling revolves around calculating causes and effects of the action potential

Definition: The membrane potential $V_m$ is the highly variable potential difference across the cell’s membrane

The key to modeling is ordinary differential equations which are solved computationally

Definition: A variable is a property of the system that changes while a parameter is a fixed property. Differential equations describe the rate of change of a variable in terms of all quantities.

The simplest and most common differential equation in this course takes the form of $\frac{d{V}_m}{dt}=G_L(E_L-V_m)/C_m$, where

• $E_L$ is the leak potential, the value that $V_m$ returns to after any temporary charge imbalance leaks away
• $G_L$ is the leak conductance, the ability of charge to leak in and out of the cell through its membrane, which increases with the surface area of the cell
• $C_m$ is the membrane capacitance, the capacity of the cell’s membrane to store electrical charge, which increases with the surface area of the cell

Cellular Background

For the most part, neurons can function as normal cells in the body. Transmembrane proteins can act as ion channels which contribute to neural function. Additional terms are relevant to neurons:

• The soma is the main body of a cell, containing the nucleus
• The process is elongated and extends far from the soma
• The dendrites are branched processes that transfer input from nearby cells/receptors to the soma
• The axons are especially thin branched processes that convey neural activity generated in the soma. They transmit action potentials, spikes in the electrical potential difference across the cell membrane.
• Vesicles are small containers of biochemical molecules, which carry neurotransmitters. These bind to neighboring neuron receptors which continues the process across multiple cells.
• Glial cells support neural activity and are less studied

On a circuit level, neurons are viewed in terms of their ramification, the amount of branching of their dendrites and axon

The complete circuit diagram of a particular region of a central nervous system, an animal’s areas of significant neural processing, is called its connectome

In some small species like C. elegans and Drosophila melanogaster, the connectome can be fully mapped out. In mammals, the connectome is more of a statistical process, coming up with a stereotypical circuit. For humans, due to scale and variability, this seems entirely unfeasible.

On a regional level, different parts of a brain will have distinct functions, shared across all members of a species

The cerebral cortex is the outer region of mammalian brains that is highly folded (with ridges/gyri and crevasses/sulci) in humans

A topographic map is an area of the brain where neurons are positioned in an organized arrangement directly related to the arrangement of their respective stimuli

Physics Background

Broadly speaking, electrical circuits can be produced whenever an electrical charge can move around. In solutions, the charge carriers are ions.

Many chemicals are soluble and can dissociate into positively and negatively charged ions. The motion of charges is mediated by an electric field, the gradient of electric potential. Positive charges are drawn to negative potential and vice versa. Each charge produces its own electric field.

An electrical current is a flow of electrical charge. In our case, $I_m=\frac{dQ}{dt}$ where $I_m$ is the inward membrane current and $Q$ is the charge on the inside of the cell’s membrane.

Voltage or potential difference is a force that moves charges. Significant potential differences arise between the inner and outer surfaces of neurons.

Capacitance (capacity to store charge) is $Q=CV$ where $Q$ is charge on a surface and $V$ is potential difference produced across that surface

Conductance (ability to let current flow) $G$ represents how current relates to potential difference. Its inverse is resistance: $V=IR$ and $I=GV$

Neurons can control voltage through ion pumps and ion channels. Notably, cytoplasm is not a particularly good conductor; different parts of a neuron will have different voltages. Models that treat a neuron as a single point lose some of this detail.

A time constant is defined by the amount of time needed for a system to get closer to equilibrium by a factor of 1/$e$. This is related to the amount of charge that is to flow and the rate of flow, which is given by the capacitance divided by the conductance.

Mathematics Background

Differential equations represent the way things change over time. This course will deal with ordinary differential equations which only describe individual properties of a system. These can be difficult to solve mathematically but we will just use computers to solve them numerically.

Example: Say the velocity of a runner is modeled with $v(t)=v_{max}(1-e^{-t/\tau })$
For a simple equation like this we can use calculus knowledge to solve $x(t)=v_{max}t-v_{max}\tau (1-e^{-t/\tau })$

Example: Sometimes a more complicated dependence exists such that velocity is a function of time and position, $\frac{dx}{dt}=v(x,t)=v_0(1+x(t{)}^2)$
Outside knowledge tells us that a solution is $x(t)=\tan (v_{0}t)$

Definition: Exponential decay occurs in a system with an equilibrium point and a rate of change proportional to distance to equilibrium

Example: Say we want to model heat entering a house, $K\frac{dT}{dt}=-A+B(T_{ext}-T)$ where $K$ is the heap capacity of the house, $A$ is the rate of heat extraction, $T_{ext}$ is the outside air temperature, and $B$ is a measure of how easily heat flows through the walls

This is both an ODE and linear, since $dT$ is proportional to $T$. The function is therefore called a linear first order ordinary differential equation.

$T_{eq}$ is when $\frac{dT}{dt}=0$
$T_{eq}=T_{ext}-A/B$
Linear first order ODE can also be written as $\frac{dT}{dt}=\frac{T_{eq}-T}{\tau }$ where $\tau$ is the time constant of the system, the inverse of the gradient. In this form it is clear that this system is stable and will tend towards equilibrium.

If the initial temperature is $T_0$ at $t=0$ then the general equation is $T(t)=T_{eq}+(T_0-T_{eq})e^{-t/\tau }$
In our example $T(t)=T_{ext}-A/B(1-e^{-Bt})$

Proof:
$\frac{dx}{dt}=\frac{x_{eq}-x}{\tau }$
$\frac{dx}{x_{eq}-x}=\frac{dt}{\tau }$
$[-\ln (x_{eq}-x){]}_{x(0)}^{x(t)}$
$\ln \left(\frac{x_{eq}-x(t)}{x_{eq}-x(0)}\right)=-t/\tau$
$\frac{x_{eq}-x(t)}{x_{eq}-x(0)}=e^{-t/\tau }$
$x(t)=x_{eq}+(x(0)-x_{eq})e^{-t/\tau }$

Example: One model in neuroscience uses gating variables $\alpha$ and $\beta$ to represent the fraction of closed ion channels opening and open ion channels closing

Say $N_O$ is the number of open channels, $N_C$ is the number of closed channels, and $N_T=N_C+N_O$ is the total. Then the rate of closed channels opening is $\alpha N_C$ and the rate of open channels closing is $\beta N_O$. Then the rate of change of the number is the opening ones minus the closing ones: $\frac{d{N}_O}{dt}=\alpha (N_T-N_O)-\beta N_O$.

The gating variable $s$ is $N_O/N_T$, so $\frac{ds}{dt}=\alpha (1-s)-\beta s$

Now $\frac{ds}{dt}=\frac{s_{eq}-s}{{\tau }_s}$ where $s_{eq}=\frac{\alpha }{\alpha +\beta }$ and ${\tau }_s=\frac{1}{\alpha +\beta }$

Therefore, $s(t)=s_{eq}+(s(0)-s_{eq})e^{-t/{\tau }_s}$

This is a good model for two-state systems. This can also be applied to multiple two-state variables as long as they are independent. As in, if multiple independent states contribute to whether a gate is open or not, then we can multiply the gating variables together to evaluate the total probability.

Linear algebra is important for large amounts of data. I just learned it so no need to recap.

Probability! The probability of an event’s occurrence ranges from $0$ to $1$:
$P(A\cup B)=P(A)+P(B)$ if $A$ and $B$ are mutually exclusive
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ if $A$ and $B$ are not mutually exclusive
$P(A\cap B)=P(A)P(B)$ if $A$ and $B$ are independent

$A$ and $B$ are independent if $P(A|B)=P(A)$ where $P(A|B)$ means “the probability of $A$ given $B$”
$A$ and $B$ are positively correlated if $P(A|B)>P(A)⟺P(B|A)>P(B)⟺P(A\cap B)>P(A)P(B)$

$P(A\cap B)=P(A)P(B|A)$ even if $A$ and $B$ are not independent
By symmetry, we find Bayes’ Theorem, $P(B|A)=\frac{P(B)P(A|B)}{P(A)}$
$P(A)$ and $P(B)$ are called priors

Example: Suppose we have five coins, four are fair and one produces heads with probability $3/4$. You choose one of the coins at random and get $3$ heads. What is the probability it is the biased coin?

$P(B|A)=\frac{(1/5)(3/4{)}^3}{(1/5)(3/4{)}^3+(4/5)(1/2{)}^3}=27/59$
Powerful conceptual idea!

In this case, the prior probability of the biased coin was $1/5$. However after flipping $3$ times, the posterior probability is $27/59\sim 0.458$. This posterior probability could be used as a prior for another experiment.

MATLAB

This stuff is not so difficult. Pretty straightforward. But the section includes some stuff on solving ODEs.

Essentially, given an ODE $\frac{dx}{dt}=f(x,t)$, one can easily write $\frac{x_{n+1}-x_n}{\Delta t}=f(x_n,t_n)$
So $x_{n+1}=x_n+f(x_n,t_n)\Delta t$
The error with this method is accurate to the order of $\Delta t/T$

Other methods, like backward Euler and Runge-Kutta methods are more accurate.

One advantage of forward Euler is how it is relatively simple to add white-noise, i.e. random variation to the system,
$\frac{dx}{dt}=f(x,t)+\sigma \cdot w(t)$ where $\sigma$ is a scaling factor for the noise
$x_{n+1}=x_n+f(x_n)\Delta t+\sigma {\stackrel{\sim }{w}}_n\sqrt{\Delta t}$, where ${\stackrel{\sim }{w}}_n$ is a random number selected from a distribution with zero mean and unit variance

Gaussian is a common choice $P({\stackrel{\sim }{w}}_n)=\frac{1}{\sqrt{2\pi }}{e}^{-{\stackrel{\sim }{w}}_n^2/2}$ and can be obtained in MATLAB easily via randn()