[Brian] Introduction to Brian part 1 : Neurons

running cell

 

from brian2 import *
%matplotlib inline # activate inline plotting

 

Units system

 

# Brian has a system for using quantities with physical dimensions

20*volt
--> 20.0V

1000*amp
--> 1.0kA

1e6*volt
--> 1.0MV

1000*namp
--> 1.0000000000000002μA

10*nA*5*Mohm
--> 49.99999999999999mV

 

A simple model

 

start_scope()

tau = 10*ms
eqs = '''
dv/dt = (1-v)/tau : 1
'''

G = NeuronGroup(1, eqs)
run(100*ms)

--> INFO       No numerical integration method specified for group 'neurongroup', using method 'exact' (took 0.02s). [brian2.stateupdaters.base.method_choice]

start_scope()

G = NeuronGroup(1, eqs, method='exact')
print('Before v = %s' % G.v[0])
run(100*ms)
print('After v = %s' % G.v[0])

--> Before v = 0.0
--> After v = 0.9999546000702376

print('Expected value of v = %s' % (1-exp(-100*ms/tau)))
--> Expected value of v = 0.9999546000702375

start_scope()

G = NeuronGroup(1, eqs, method='exact')
M = StateMonitor(G, 'v', record=True)

run(30*ms)

plot(M.t/ms, M.v[0])
xlabel('Time (ms)')
ylabel('v');

start_scope()

G = NeuronGroup(1, eqs, method='exact')
M = StateMonitor(G, 'v', record=0)

run(30*ms)

plot(M.t/ms, M.v[0], 'C0', label='Brian')
plot(M.t/ms, 1-exp(-M.t/tau), 'C1--',label='Analytic')
xlabel('Time (ms)')
ylabel('v')
legend();

start_scope()

tau = 10*ms
eqs = '''
dv/dt = (sin(2*pi*100*Hz*t)-v)/tau : 1
'''

# Change to Euler method because exact integrator doesn't work here
G = NeuronGroup(1, eqs, method='euler')
M = StateMonitor(G, 'v', record=0)

G.v = 5 # initial value

run(60*ms)

plot(M.t/ms, M.v[0])
xlabel('Time (ms)')
ylabel('v');

 

Adding spikes

 

start_scope()

tau = 10*ms
eqs = '''
dv/dt = (1-v)/tau : 1
'''

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v = 0', method='exact')

M = StateMonitor(G, 'v', record=0)
run(50*ms)
plot(M.t/ms, M.v[0])
xlabel('Time (ms)')
ylabel('v');

start_scope()

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v = 0', method='exact')

spikemon = SpikeMonitor(G)

run(50*ms)

print('Spike times: %s' % spikemon.t[:])
--> Spike times: [16.  32.1 48.2] ms

start_scope()

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v = 0', method='exact')

statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='C1', lw=3)
xlabel('Time (ms)')
ylabel('v');

 

Refractoriness

 

start_scope()

tau = 10*ms
eqs = '''
dv/dt = (1-v)/tau : 1 (unless refractory)
'''

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v = 0', refractory=5*ms, method='exact')

statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='C1', lw=3)
xlabel('Time (ms)')
ylabel('v');

start_scope()

tau = 5*ms
eqs = '''
dv/dt = (1-v)/tau : 1
'''

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v = 0', refractory=15*ms, method='exact')

statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='C1', lw=3)
axhline(0.8, ls=':', c='C2', lw=3)
xlabel('Time (ms)')
ylabel('v')
print("Spike times: %s" % spikemon.t[:])
--> Spike times: [ 8. 23. 38.] ms

 

Multiple neurons

 

start_scope()

N = 100
tau = 10*ms
eqs = '''
dv/dt = (2-v)/tau : 1
'''

G = NeuronGroup(N, eqs, threshold='v>1', reset='v=0', method='exact')
G.v = 'rand()'

spikemon = SpikeMonitor(G)

run(50*ms)

plot(spikemon.t/ms, spikemon.i, '.k')
xlabel('Time (ms)')
ylabel('Neuron index');

 

Parameters

 

start_scope()

N = 100
tau = 10*ms
v0_max = 3.
duration = 1000*ms

eqs = '''
dv/dt = (v0-v)/tau : 1 (unless refractory)
v0 : 1
'''

G = NeuronGroup(N, eqs, threshold='v>1', reset='v=0', refractory=5*ms, method='exact')
M = SpikeMonitor(G)

G.v0 = 'i*v0_max/(N-1)'

run(duration)

figure(figsize=(12,4))
subplot(121)
plot(M.t/ms, M.i, '.k')
xlabel('Time (ms)')
ylabel('Neuron index')
subplot(122)
plot(G.v0, M.count/duration)
xlabel('v0')
ylabel('Firing rate (sp/s)');

 

Stochastic neurons

 

start_scope()

N = 100
tau = 10*ms
v0_max = 3.
duration = 1000*ms
sigma = 0.2

eqs = '''
dv/dt = (v0-v)/tau+sigma*xi*tau**-0.5 : 1 (unless refractory)
v0 : 1
'''

G = NeuronGroup(N, eqs, threshold='v>1', reset='v=0', refractory=5*ms, method='euler')
M = SpikeMonitor(G)

G.v0 = 'i*v0_max/(N-1)'

run(duration)

figure(figsize=(12,4))
subplot(121)
plot(M.t/ms, M.i, '.k')
xlabel('Time (ms)')
ylabel('Neuron index')
subplot(122)
plot(G.v0, M.count/duration)
xlabel('v0')
ylabel('Firing rate (sp/s)');

start_scope()

N = 1000
tau = 10*ms
vr = -70*mV
vt0 = -50*mV
delta_vt0 = 5*mV
tau_t = 100*ms
sigma = 0.5*(vt0-vr)
v_drive = 2*(vt0-vr)
duration = 100*ms

eqs = '''
dv/dt = (v_drive+vr-v)/tau + sigma*xi*tau**-0.5 : volt
dvt/dt = (vt0-vt)/tau_t : volt
'''

reset = '''
v = vr
vt += delta_vt0
'''

G = NeuronGroup(N, eqs, threshold='v>vt', reset=reset, refractory=5*ms, method='euler')
spikemon = SpikeMonitor(G)

G.v = 'rand()*(vt0-vr)+vr'
G.vt = vt0

run(duration)

_ = hist(spikemon.t/ms, 100, histtype='stepfilled', facecolor='k', weights=list(ones(len(spikemon))/(N*defaultclock.dt)))
xlabel('Time (ms)')
ylabel('Instantaneous firing rate (sp/s)');

 

https://brian2.readthedocs.io/en/stable/resources/tutorials/1-intro-to-brian-neurons.html

Introduction to Brian part 1: Neurons — Brian 2 2.5.0.3 documentation