Pitt MATH 3370 Mathematical Neuroscience Models

An collection of models for use in Pitt Math 3370 Mathematical Neuroscience. 🧠

Installation

How do you install this package?

Dependencies

Numpy, Scipy, maybe a C compiler if I get around to the stochastic coupling

Supported models

Hodgkin-Huxley

4D model in $V, m, h, n$. Default initial conditions are $(V, m, h, n) = (-65, 0.05, 0.6, 0.317)$.

The dynamics of the model are given by

$$ c\frac{dV}{dt} = I_0 + I_{\text{inj}} - I_{\text{Na}} - I_{\text{K}} - g_{\text{Leak}}(V - V_{\text{Leak}}), $$

$$ \frac{dm}{dt} = \alpha_{m}(V)(1 - m) - \beta_m(V)m, $$

$$ \frac{dh}{dt} = \alpha_{h}(V)(1 - h) - \beta_h(V)h, $$

$$ \frac{dn}{dt} = \alpha_{n}(V)(1 - n) - \beta_n(V)n, $$

with the $\alpha$ and $\beta$ given by

$$ \alpha_m(V) = \phi \times .1 \times \frac{(V + 40)}{1 - \exp\left[-(V + 40)/10\right]} \qquad \beta_m(V) = \phi\times 4 \exp\left[-(V + 65)/18\right] $$

$$ \alpha_h(V) = \phi \times 0.07\exp\left[-(V + 65)/20\right] \qquad \beta_h(V) = \phi\frac 1 {1 + \exp\left[-(V + 35)/10\right]} $$

$$ \alpha_n(V) = \phi\times 0.01 \frac{V + 55}{\exp\left[-(V + 55)/10\right]} \qquad \beta_n(V) = \phi\times0.125\exp\left[-(V + 65)/80\right]. $$

Finally, the currents are given by $$ I_{\text{Na}} = g_{\text{Na}}h(V - V_{\text{Na}}) m^3, \quad I_{\text{K}} = g_{\text{K}} (V - V_{\text{K}}) n^4. $$

These are default values for parameterizing the $\alpha$ and $\beta$.

TBD: should these be expressed in a more generic way?

The reversal potentials and conductances default to

$$ V_{\text{Na}} = 50 \quad V_{\text{K}} = -77 \quad V_{\text{Leak}} = -54.387 $$

$$ g_{\text{Na}} = 120 \quad g_{\text{K}} = 36 \quad g_{\text{Leak}} = 0.3. $$

$c$ and $\phi$ default to 1.

Rinzel Reduction

The Rinzel reduction of the HH model inherits the $\alpha$ and $\beta$ functions, as well as the reversal potentials, channel conductances, $c$ and $\phi$. The dynamics for $V$ and $n$ are also inherited, but $h$ and $m$ are modeled:

$$ \frac{dm}{dt} = \frac{\alpha_{m}(V)}{\alpha_m(V) + \beta_m(V)}, $$

$$ h = h_0 - n. $$

$h_0$ is a constant that defaults to 0.8.

Kepler Reduction

Where did I get these equations from? Canvas?

Integrate-and-fire

Quadratic

Destexhe-Pare

Destexhe-Pare is a 5D model in $V, m, h, n$ and $m_{\text{K}}$, with default initial conditions

$$ (V, m, h, n, m_{\text{K}}) = (-73.87,0,1,0.002,0.0075). $$

The dynamics of $m, n,$ and $h$ are the same as in HH, with the dynamics of $V$ and $m_{\text{K}}$ given by:

$$ c\frac{dV}{dt} = I_0 + I_{\text{Inj}} - g_{\text{L}}(V - E_{\text{L}}) - I_{\text{Kdr}}(V,n) - I_{\text{Na}}(V, m, h) - I_{\text{Km}} $$

$$ \frac{dm_{\text{K}}}{dt} = \alpha_{m_{\text{K}}}(V)(1 - m_{\text{K}}) - \beta_{m_{\text{K}}}(V)m_{\text{K}}. $$

The sodium current is the same as HH, and the $I_{\text{Kdr}}$ of this model is the $I_{\text{K}}$ of HH. The new current is

$$ I_{\text{Km}}(V,m)= g_{\text{Km}}m(V-e_{\text{K}}). $$

Finally, the $\alpha$ and $\beta$ take on different forms:

$$ \alpha_m(V)=-.32\frac{V-V_t-13}{\exp(-(V-V_t-13)/4)-1)} \quad \beta_m(V)=.28\frac{V-V_t-40}{\exp((V-V_t-40)/5)-1)} $$

$$ \alpha_h(V)=.128\exp(-(V-V_t-V_s-17)/18) \quad \beta_h(V)=\frac 4 {1+\exp(-(V-V_t-V_s-40)/5)} $$

$$ \alpha_n(V)=-.032\frac{V-V_t-15}{\exp(-(V-V_t-15)/5)-1} \quad \beta_n(V)=.5\frac 1 {\exp(-(V-V_t-10)/40)} $$

$$ \alpha_{text{Km}}(V)= .0001\frac{V+30}{1-\exp(-(V+30)/9)} \quad \beta_{\text{Km}}(V)=-.0001\frac{V+30}{1-\exp((V+30)/9)} $$

Izhikevic

The Izhikevic model is 2D in $u$ and $V$, with dynamics

$$ \dot V = I_0 + I_{\text{inj}} + V^2 - u, $$

$$ \dot u = a(bV - u). $$

On $V$ crossing $V_{\text{threshold}}$ from below, the resetting

$$ V\to c, u \to u + d $$

takes place. $V_{\text{threshold}}$, $a, b, c$, and $d$ are all adjustable member variables of the model.

TODO: Need reasonable defaults.

Morris-Lecar

How to couple models

How do you couple models?

Other functionality

Gillespie Simulations

Currently supports