1. Dan Goodman & Romain Brette Ecole Normale Supérieure Projet Odyssée http://brian.di.ens.fr goodman@di.ens.fr brette@di.ens.fr

2. Structure of a neuron Dendrites Axon Synapse Soma (cell body)

3. Membrane potential Inside cellOutside cellMembrane (semi-permeable) Potential difference (V) The difference in concentrations of sodium and potassium ions (mostly). Roughly -70 mV at rest

4. Action potentials (aka spikes)

5. Synapses Presynaptic terminal Postsynaptic terminal Synaptic cleft Neurotransmitter

6. Model neurons “Single compartment model” – the simplest Time evolution (differential equation) Spike propagation (delta function) Spike initiation – Threshold condition – Action potential (spike) – Reset V

7. 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

8. 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

9. Introducing Brian – leaky I&F Code is in Python Equations (differential, define time evolution) Threshold Reset Model NeuronGroup Connection

10. Brian is flexible Threshold increases when spike arrives and decays Implemented as DE and user-defined reset and threshold functions

11. 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

12. Data structures State matrix S, values of model variables at any given time Update matrix A Encodes exact solution to linear differential equation V=0 x=1 y=2 Weight matrix W, synapse strengths

13. Brian’s vector operations Time evolution S=dot(A,S) Threshold spikes=(S[0,:]>Vt).nonzero()[0] Spike propagation S[0,:]+=W[spikes,:] Reset S[0,spikes]=Vr * State matrix SUpdate matrix A V x y

14. 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