[Neuronal dynamics] synaptic currents simulation with python

3.1 Synapses

Goal: make below GABAa current with python

target figure

GABAa Current symulation

1. equation of synaptic current. I_syn(t)

t : time.

gsyn(t) : synaptic conductance. Here, g_syn

u(t) : resting membrane potential. Here, E_memb

Esyn : reversal potential. Here, E_rev

def I_syn(t, g_syn, E_memb, E_rev):
    return g_syn(t)*(E_memb-E_rev)

Nothing difficult. Exact same function written from formula.

2. equation of synaptic conductance decay. g_syn(t)

g`_syn : amplitude.

t^(f) : arrival time of action potential

tau : time constant of rise time.

theta(t-t^(f)) : Heavyside step function. If t^(f)equals 0, it means event is happened at 0.

def create_g_syn(t, _g_syn=40, tf=0, tau=5):
    def g_syn(t):
        return _g_syn*np.exp(-(t-tf)/tau)*np.heaviside(t-tf, 1)
    return g_syn

With numpy module. np.exp and np.heaviside usede here.

For heaviside function H(x),

second argument 0.5 is indicating value of 1/2 if x=0 .

_g_syn is amplitude

here, defining function inside the function, we can flexibly change constants of _g_syn , tf , tau .

3. equation of synaptic conductance rise and decay. g_syn2(t)

First [ ] shows rise,

Second [ ] shows decay, composed with fast and slow components.

if there is no rise element and a=1, than it is similar with g_syn(t) in part 2.

def create_g_syn2(t, _g_syn=40, tf=0, tau=5, tau_rise=1, a=1, tau_fast=6, tau_slow=100):
    def g_syn2(t):
        rise = 1-np.exp(-(t-tf)/tau_rise)
        fast = a*np.exp(-(t-tf)/tau_fast)
        slow = (1-a)*np.exp(-(t-tf)/tau_slow)
        return _g_syn*rise*(fast+slow)*np.heaviside(t-tf, 0.5)
    return g_syn2

this is creates g_syn2 function.

4. Finding constants in text.

For constant values,

E_rev = -75

_g_syn=40

tf=0, E_memb = -65

a = 1, tau_rise = 1, tau_fast = 6, tau_slow = ?? (it’s ok because a=1, which removes tau_slow part)

5. plot and comparison of decay model and rise&decay model.

t = np.arange(-10,20, 0.1)
g_syn = create_g_syn(t, _g_syn=40, tf=0, tau=5)
y = -I_syn(t, g_syn=g_syn, E_memb=-65, E_rev=-70)
g_syn2 = create_g_syn2(t, _g_syn=40, tf=0, tau=5)
y2 = -I_syn(t, g_syn=g_syn2, E_memb=-65, E_rev=-70)
plt.plot(t, y)
plt.plot(t, y2)

g_syn(t) [blue: only decay] and g_syn2(t) [orange: rise + decay]

6. Full code. GABAa synaptic conductance!

goal achieved!

import numpy as np
import matplotlib.pyplot as plt
def I_syn(t, g_syn, E_memb, E_rev):
    return g_syn(t)*(E_memb-E_rev)
def create_g_syn(t, _g_syn=40, tf=0, tau=5):
    def g_syn(t):
        return _g_syn*np.exp(-(t-tf)/tau)*np.heaviside(t-tf, 0.5)
    return g_syn
def create_g_syn2(t, _g_syn=40, tf=0, tau=5, tau_rise=1, a=1, tau_fast=6, tau_slow=100):
    def g_syn2(t):
        rise = 1-np.exp(-(t-tf)/tau_rise)
        fast = a*np.exp(-(t-tf)/tau_fast)
        slow = (1-a)*np.exp(-(t-tf)/tau_slow)
        return _g_syn*rise*(fast+slow)*np.heaviside(t-tf, 0.5)
    return g_syn2
t = np.arange(-10,500, 0.1)
# g_syn = create_g_syn(t, _g_syn=40, tf=0, tau=5)
# y = -I_syn(t, g_syn=g_syn, E_memb=-65, E_rev=-70)
g_syn2 = create_g_syn2(t, _g_syn=40, tf=0, tau=5)
y2 = -I_syn(t, g_syn=g_syn2, E_memb=-65, E_rev=-70)
# plt.plot(t, y)
plt.plot(t, y2)
plt.xlim(-10, 500)
plt.ylim(-300, 400)

Written for study purpose.