Simulating neural networks with Romain Brette ([email protected])

Simulating neural networks with

Romain Brette ([email protected])

SPIKING NEURON MODELS

Spiking neuron models

Input =N spike trains

Output =1 spike train

A neuron model is defined by:• what happens when a spike is received• the condition for spiking• what happens when a spike is produced• what happens between spikes

discrete events

continuous dynamics

The membrane equation

Iinj

outside

inside

injLmm I

R

EV

dt

dVC

)(

=1/R

Vm

injmLm RIVEdt

dV RC membrane time constant

(typically 3-100 ms)

Spiking:If V = Vt (threshold)then: neuron spikes and V→Vr (reset)

The Hodgkin-Huxley model

Hodgkin & Huxley(Nobel prize 1963)

nVndt

dnV

hVhdt

dhV

mVmdt

dmV

VEngVEhmgVEgdt

dVC

n

h

m

KKNall

)()(

)()(

)()(

)()()( 43

Model of the space clamped giant squid axon

Synaptic integration

spike threshold

action potential or « spike »

postsynaptic potential (PSP)

temporal integration

Synaptic currentssynaptic current

postsynaptic neuron

synapse

Is(t)

smLm RIVEdt

dV

Idealized synapse• Total charge• Opens for a short duration• Vm increases by Q/C

sIQ

EL

t

Lm eRQ

EtV

)(

at t=0

A more realistic synapse model)( msss VEgI

ionic channel conductance

synaptic reversal potential

gs(t)

presynaptic spike

open closed

))(( mssmLm VEtRgVEdt

dV

Example of kinetic model• Stochastic transitions between open

and closed

C ⇄ Oα[L]

βopening rate, proportional to concentration

constant closing rate

Macroscopic equation (many channels):

xxLdt

dx )1]([ proportion of open channels

Assuming neurotransmitter are present for a very short duration:

ss

s gdt

dg

τs=1/β

gs(t)=x(t)*gmax

ss gg

General spiking neuron models

)(:

)(

)(

XrXSX

XgX

Xfdt

dX

i

presynaptic spike(synapse i)

threshold reset

evolution of state variables

)(:

)(

)(

XrXSX

XgX

Xfdt

dX

i

)(:

)(

)(

XrXSX

XgX

Xfdt

dX

i

)(:

)(

)(

XrXSX

XgX

Xfdt

dX

i

transmission delayTypically: V=Vt: V→Vr

http://www.briansimulator.org

The spirit of “A simulator should not only save

the time of processors, but also the time of scientists”

scientist computer

Writing code often takes more time than running it

Goals:• Quick model coding• Flexible

models are defined by equations (rather than pre-defined)

Goodman, D. and R. Brette (2009). The Brian simulator. Front Neurosci doi:10.3389/neuro.01.026.2009.

Who is Brian?

briansimulator.org

Example: current-frequency curvefrom brian import *

N = 1000tau = 10 * mseqs = '''dv/dt=(v0-v)/tau : voltv0 : volt'''group = NeuronGroup(N, model=eqs, threshold=10 * mV, reset=0 * mV, refractory=5 * ms)group.v = 0 * mVgroup.v0 = linspace(0 * mV, 20 * mV, N)

counter = SpikeCounter(group)

duration = 5 * secondrun(duration)plot(group.v0 / mV, counter.count / duration)show()

Example: a fully connected networkfrom brian import *

tau = 10 * msv0 = 11 * mVN = 20w = .1 * mV

group = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)

W = Connection(group, group, 'v', weight=w)

group.v = rand(N) * 10 * mV

S = SpikeMonitor(group)

run(300 * ms)

raster_plot(S)show()

Example: a fully connected networkfrom brian import *

tau = 10 * msv0 = 11 * mVN = 20w = .1 * mV

group = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)

W = Connection(group, group, 'v', weight=w)

group.v = rand(N) * 10 * mV

S = SpikeMonitor(group)

run(300 * ms)


R.E. Mirollo and S.H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM Journal on Applied Mathematics 50, 1645-1662 (1990).

Example 2: a ring of IF neuronsfrom brian import *

tau = 10 * msv0 = 11 * mVN = 20w = 1 * mV

ring = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)

W = Connection(ring, ring, 'v')for i in range(N): W[i, (i + 1) % N] = w

ring.v = rand(N) * 10 * mV

S = SpikeMonitor(ring)

run(300 * ms)


Example 2: a ring of IF neuronsfrom brian import *

tau = 10 * msv0 = 11 * mVN = 20w = 1 * mV

ring = NeuronGroup(N, model='dv/dt=(v0-v)/tau : volt', threshold=10 * mV, reset=0 * mV)

W = Connection(ring, ring, 'v')for i in range(N): W[i, (i + 1) % N] = w

ring.v = rand(N) * 10 * mV

S = SpikeMonitor(ring)

run(300 * ms)


Example 3: a noisy ringfrom brian import *

tau = 10 * mssigma = .5N = 100mu = 2

eqs = """dv/dt=mu/tau+sigma/tau**.5*xi : 1"""

group = NeuronGroup(N, model=eqs, threshold=1, reset=0)

C = Connection(group, group, 'v')for i in range(N): C[i, (i + 1) % N] = -1

S = SpikeMonitor(group)trace = StateMonitor(group, 'v', record=True)

run(500 * ms)subplot(211)raster_plot(S)subplot(212)plot(trace.times / ms, trace[0])show()

Example 3: a noisy ringfrom brian import *

tau = 10 * mssigma = .5N = 100mu = 2

eqs = """dv/dt=mu/tau+sigma/tau**.5*xi : 1"""

group = NeuronGroup(N, model=eqs, threshold=1, reset=0)

C = Connection(group, group, 'v')for i in range(N): C[i, (i + 1) % N] = -1

S = SpikeMonitor(group)trace = StateMonitor(group, 'v', record=True)

run(500 * ms)subplot(211)raster_plot(S)subplot(212)plot(trace.times / ms, trace[0])show()

Example 4: topographic connectionsfrom brian import *

tau = 10 * msN = 100v0 = 5 * mVsigma = 4 * mVgroup = NeuronGroup(N, model='dv/dt=(v0-v)/tau + sigma*xi/tau**.5 : volt', threshold=10 * mV, reset=0 * mV)C = Connection(group, group, 'v', weight=lambda i, j:.4 * mV * cos(2. * pi * (i - j) * 1. / N))S = SpikeMonitor(group)R = PopulationRateMonitor(group)group.v = rand(N) * 10 * mV

run(5000 * ms)subplot(211)raster_plot(S)subplot(223)imshow(C.W.todense(), interpolation='nearest')title('Synaptic connections')subplot(224)plot(R.times / ms, R.smooth_rate(2 * ms, filter='flat'))title('Firing rate')show()

Example 4: topographic connectionsfrom brian import *

tau = 10 * msN = 100v0 = 5 * mVsigma = 4 * mVgroup = NeuronGroup(N, model='dv/dt=(v0-v)/tau + sigma*xi/tau**.5 : volt', threshold=10 * mV, reset=0 * mV)C = Connection(group, group, 'v', weight=lambda i, j:.4 * mV * cos(2. * pi * (i - j) * 1. / N))S = SpikeMonitor(group)R = PopulationRateMonitor(group)group.v = rand(N) * 10 * mV

run(5000 * ms)subplot(211)raster_plot(S)subplot(223)imshow(C.W.todense(), interpolation='nearest')title('Synaptic connections')subplot(224)plot(R.times / ms, R.smooth_rate(2 * ms, filter='flat'))title('Firing rate')show()

Example 5: random networkfrom brian import *

taum = 20 * msecondtaue = 5 * msecondtaui = 10 * msecondEe = 60 * mvoltEi = -80 * mvolt

eqs = '''dv/dt = (-v+ge*(Ee-v)+gi*(Ei-v))*(1./taum) : voltdge/dt = -ge/taue : 1dgi/dt = -gi/taui : 1'''

P = NeuronGroup(4000, model=eqs, threshold=10 * mvolt, reset=0 * mvolt, refractory=5 * msecond)Pe = P.subgroup(3200)Pi = P.subgroup(800)we = 0.6wi = 6.7Ce = Connection(Pe, P, 'ge', weight=we, sparseness=0.02)Ci = Connection(Pi, P, 'gi', weight=wi, sparseness=0.02)P.v = (randn(len(P)) * 5 - 5) * mvoltP.ge = randn(len(P)) * 1.5 + 4P.gi = randn(len(P)) * 12 + 20

S = SpikeMonitor(P)

run(1 * second)raster_plot(S)show()

Example 5: random networkfrom brian import *

taum = 20 * msecondtaue = 5 * msecondtaui = 10 * msecondEe = 60 * mvoltEi = -80 * mvolt

eqs = '''dv/dt = (-v+ge*(Ee-v)+gi*(Ei-v))*(1./taum) : voltdge/dt = -ge/taue : 1dgi/dt = -gi/taui : 1'''

P = NeuronGroup(4000, model=eqs, threshold=10 * mvolt, reset=0 * mvolt, refractory=5 * msecond)Pe = P.subgroup(3200)Pi = P.subgroup(800)we = 0.6wi = 6.7Ce = Connection(Pe, P, 'ge', weight=we, sparseness=0.02)Ci = Connection(Pi, P, 'gi', weight=wi, sparseness=0.02)P.v = (randn(len(P)) * 5 - 5) * mvoltP.ge = randn(len(P)) * 1.5 + 4P.gi = randn(len(P)) * 12 + 20

S = SpikeMonitor(P)

run(1 * second)raster_plot(S)show()

Synaptic plasticity

Presynaptic spike

Post

syna

ptic

spik

e

Dan

& P

oo (2

006)

pre post: potentiation post pre: depression

(STDP = Spike-Timing-Dependent Plasticity)

Phenomenological model

postpost

post

prepre

pre

Adt

dA

Adt

dA

Presynaptic spike:

post

preprepre

Aww

AAA

Postsynaptic spike:

pre

postpostpost

Aww

AAA

Synaptic plasticity with Brian

synapses=Connection(input,neurons,'ge') eqs_stdp='''dA_pre/dt=-A_pre/tau_pre : 1dA_post/dt=-A_post/tau_post : 1''‘stdp=STDP(synapses,eqs=eqs_stdp,pre='A_pre+=dA_pre;w+=A_post', post='A_post+=dA_post;w+=A_pre',wmax=gmax)

maximum weight

The Synapses classS=Synapses(source,target,model="""w : 1 p : 1""", pre="v+=w*(rand()<p)")# probabilistic synapse

S = Synapses(source, target, '''w:1 dApre/dt=-Apre/taupre : 1 dApost/dt=-Apost/taupost : 1''', pre='''ge+=w Apre+=dApre w=clip(w+Apost,0,gmax)''', post='''Apost+=dApost w=clip(w+Apre,0,gmax)'’’)

https://github.com/brian-team/brian2http://brian2.readthedocs.org

What’s new with ?

• More flexibility• Code is generated for targets

Python C++ Android Anything else

Achilles Koutsou

BrianDROID

Thanks

Bertrand Fontaine Cyrille Rossant

Dan Goodman

Victor Benichoux

Marcel Stimberg

http://www.briansimulator.org

Documents

Simulating neural networks with Romain Brette ([email protected])