http://brian.di.ens.fr
Dan Goodman & Romain Brette
Ecole Normale Supérieure
Projet Odyssée
goodman@di.ens.fr
brette@di.ens.fr
Structure of a neuron
Synapse
Axon
Dendrites
Soma
(cell body)
Membrane potential
Potential difference (V)
The difference in concentrations of
sodium and potassium ions
(mostly).
Roughly -70 mV at rest
Inside cell
Membrane
(semi-permeable)
Outside cell
Action potentials (aka spikes)
Synapses
Neurotransmitter
Presynaptic terminal
Postsynaptic
terminal
Synaptic cleft
Model neurons
•
•
•
•
“Single compartment model” – the simplest
Time evolution (differential equation)
Spike propagation (delta function)
Spike initiation
– Threshold condition
– Action potential (spike)
– Reset
V
Integrate and Fire model
• One variable V
• No time evolution, or
dV/dt=0
• Spike propagation,
when spike arrives set
V→V+w
• Threshold, fire spike if
V>Vt
• Reset, after spike V→Vr
Leaky I&F
• One variable V
• Exponential decay (‘leak
current’) or
τ dV/dt = -(V-Vr)
• Spike propagation,
when spike arrives set
V→V+w
• Threshold, fire spike if
V>Vt
• Reset, after spike V→Vr
Introducing Brian – leaky I&F
• Code is in Python
• Equations (differential,
define time evolution)
• Threshold
• Reset
• Model
• NeuronGroup
• Connection
Brian is flexible
• Threshold increases when
spike arrives and decays
• Implemented as DE and
user-defined reset and
threshold functions
Efficiency: vectorisation
• Python is slow (interpreted)
• In Brian most operations are vector operations
(same operation on multiple pieces of data)
• Use NumPy for vector operations
• Linear differential equation, use matrix
algebra for exact update t→t+dt
• Spike propagation, V→V+w for certain V, w
• Threshold, V>Vt
• Reset V→Vr
Data structures
State matrix S, values of model variables at any given time
V=0
x=1
y=2
Weight matrix W, synapse strengths
Update matrix A
Encodes exact solution to linear
differential equation
Brian’s vector operations
Update matrix A
State matrix S
V
x
y
Time evolution
S=dot(A,S)
*
Threshold
spikes=(S[0,:]>Vt).nonzero()[0]
Spike propagation
S[0,:]+=W[spikes,:]
Reset
S[0,spikes]=Vr
The End
• Brian is useful for modelling if:
– Network of spiking neurons
– Each neuron is modelled as single compartment
– Not too many neurons (tens of thousands?)
• Benefits are:
– Easy to learn and use compared to other software
– Quick to implement and tweak models