Pitt MATH 3370 Mathematical Neuroscience Models An collection of models for use in Pitt Math 3370 Mathematical Neuroscience. 🧠
How do you install this package?
Numpy, Scipy, maybe a C compiler if I get around to the stochastic coupling
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.
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.
Where did I get these equations from? Canvas?
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:
$$
\alpha_m(V)=-.32\frac{V-V_t-13}{\exp(-(V-V_t-13)/4)-1)}
\quad
\beta_m(V)=.28\frac{V-V_t-40}{\exp((V-V_t-40)/5)-1)}
$$
$$
\alpha_h(V)=.128\exp(-(V-V_t-V_s-17)/18)
\quad
\beta_h(V)=\frac 4 {1+\exp(-(V-V_t-V_s-40)/5)}
$$
$$
\alpha_n(V)=-.032\frac{V-V_t-15}{\exp(-(V-V_t-15)/5)-1}
\quad
\beta_n(V)=.5\frac 1 {\exp(-(V-V_t-10)/40)}
$$
$$
\alpha_{text{Km}}(V)= .0001\frac{V+30}{1-\exp(-(V+30)/9)}
\quad
\beta_{\text{Km}}(V)=-.0001\frac{V+30}{1-\exp((V+30)/9)}
$$
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.
How do you couple models?
Currently supports