1. Romain Brette & Dan Goodman Ecole Normale Supérieure Equipe Audition http://www.briansimulator.org romain.brette@ens.fr dan.goodman@ens.fr

2. 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)

3. An example PiPe Ce Ci P briansimulator.org

4. briansimulator.org, click Download We have Windows versions on USB sticks (ask Dan) Installation Download slides and examples: click Blog and Brian at NeuroComp

5. does an adaptive threshold from brian import * eqs = ''' dv/dt = -v/(10*ms) : volt dvt/dt = (10*mV-vt)/(15*ms) : volt ''' reset = ''' v = 0*mV vt += 3*mV ''' IF = NeuronGroup(1, eqs, reset=reset, threshold='v>vt') IF.rest() PG = PoissonGroup(1, 500*Hz) C = Connection(PG, IF, 'v', weight=3*mV) run(100*ms)

6. does synaptic plasticity Song, S., K. D. Miller, and L. F. Abbott (2000). Competitive hebbian learning through spike timing-dependent synaptic plasticity. Nature Neurosci 3, 919-26.

7. does systems neuroscience Sturzl, W., R. Kempter, and J. L. van Hemmen (2000). Theory of arachnid prey localization. Physical Review Letters 84 (24), 5668

8. Work in progress  Parallel computing with graphics processing units (GPUs)  Up to 240 processors  Up to 100x speed improvement  Cheap hardware (a few €100)  Good for vectorised code If you want to contribute: http://groups.google.fr/group/brian-on-gpu

9. Python in 15 minutes see also: http://docs.python.org/tutorial/http://docs.python.org/tutorial/

10. The Python console  On Windows, open IDLE. On Linux: type python  interpreted language  dynamic typing  garbage collector  space matters (signals structure)  object-oriented  many libraries

11. Writing a script  If you use IDLE, click on File>New window. Otherwise use any text editor. Press F5 to execute

12. A simple program # Factorial function def factorial(x): if x == 0: return 1 else: return x * factorial(x-1) print factorial(5) comment function definition untyped argument condition function call display structure by indentation (block = aligned instructions)

13. Numerical objects  Base types: int, long, float, complex  Other numerical types (vectors, matrices) defined in the Numpy library (in a few minutes) x=3+2 x+=1 y=100L z=x*(1+2j) u=2.3/7 x,y,z = 1,2,3 a = b = 123

14. Control structures x = 12 if x 10 and x < 20): print "Good value." if x < 5 or 10 < x < 20: print ‘Good value as well.' for i in [0,1,2,3,4]: print "Number", i for i in range(5): print "Number", i while x >= 0: print "x is not always negative." x = x-1 list the same list

15. Lists mylist = [1,7,8,3] name = ["Jean","Michel"] x = [1,2,3,[7,8],"fin"] print name[0] name[1]="Robert" print mylist[1:3] print mylist[:3],mylist[:] print mylist[-1] print x[3][1] name.append("Georges") print mylist+name print mylist*2 concatenate heterogeneous list first element = index 0 « slice »: index 3 not included last element x[3] is a list method (list = object) Other methods: extend, insert, reverse, sort, remove…

16. List comprehensions carres=[x**2 for i in range(10)] pairs=[x for i in range(10) if i % 2 == 0] = list of squares of integers from 0 to 9 = list of even integers between 0 and 9

17. Strings a="Bonjour" b='hello' c=""" Une phrase qui n'en finit pas """ print a+b print a[3:7] print b.capitalize() multiline string ≈ list of characters many methods (find, replace, split…)

18. Dictionaries dico={‘one':1,‘two':2,‘three':‘several'} print dico[‘three'] dico[‘four']=‘many‘ del dico[‘one'] keyvalue for key in dico: print key,'->',dico[key] iterate all keys

19. Functions def power(x,exponent=2): return x**exponent print power(3,2) print power(7) print power(exponent=3,x=2) carre=lambda x:x**2 inline definition default value call with named arguments

20. Modules import math print math.exp(1) from math import exp print exp(1) from math import * print exp(1) import only the exp object import everything You can work with several files (= modules), each one can define any number of objects (= variables, functions, classes) loads the file ‘math.py’ or ‘math.pyc’ (compiled)

21. Scipy & Pylab

22. Scipy & Numpy  Scientific libraries  Syntax ≈ Matlab  Many mathematical functions from scipy import * x=array([1,2,3]) M=array([[1,2,3],[4,5,6]]) M=ones((3,3)) z=2*x y=dot(M,x) from scipy.optimize import * print fsolve(lambda x:(x-1)*(x-3),2) vector matrix matrix product

23. Vectors et matrices  Base type in SciPy: array (= vector or matrix) from scipy import * x=array([1,2,3]) M=array([[1,2,3],[4,5,6]]) M=ones((3,2)) z=2*x+1 y=dot(M,x) vector (1,2,3) matrix 1 2 3 4 5 6 matrix 1 matrix product

24. Operations x+y x-y x*y x/y x**2 exp(x) sqrt(x) dot(x,y) dot(M,x) M.T M.max() M.sum() size(x) M.shape element-wise dot product matrix product transpose x² total number of elements

25. Indexing x[i] M[i,j] x[i:j] M[i,:] M[:,i] x[[1,3]] x[1:3]=[0,1] M[1,:]+=x Vector indexing  lists (first element= 0) (i+1) th element slice from x[i] to x[j-1] (i+1) th row (i+1) th column elements x[1] and x[3] x[1]=0, x[3]=1 add vector x to the 2 nd row of M M[i,:] is a « view » on matrix M  copy (  reference) y=M[0,:] y[2]=5 x=z x[1]=3 M[0,2] is 5 z[1] is 3 x=z.copy() copy:

26. Construction x=array([1,2,3]) M=array([[1,2,3],[4,5,6]]) x=ones(5) M=zeros((3,2)) M=eye(3) M=diag([1,3,7]) x=rand(5) x=randn(5) x=arange(10) x=linspace(0,1,100) from lists vector of 1s zero matrix identity matrix diagonal matrix random vector in (0,1) random gaussian vector 0,1,2,...,9 100 numbers between 0 and 1

27. SciPy example: optimisation  Simple example, least squares polynomial fit

28. Vectorisation  How to write efficient programs?  Replace loops by vector operations for i in range(1000000): X[i]=1 X=ones(1000000) for i in range(1000000): X[i]=X[i]*2 X=X*2 for i in range(999999): Y[i]=X[i+1]-X[i] Y=X[1:]-X[:-1] for i in range(1000000): if X[i]>0.5: Y[i]=1 Y[X>.5]=1

29. Pylab

30.  Plotting library  Syntax ≈ Matlab  Many plotting functions from pylab import * plot([1,2,3],[4,5,6]) show() x y last instruction of script plot polar more examples: http://matplotlib.sourceforge.net/gallery.html

31. Plotting with PyLab hist(randn(1000))contour(gaussian_filter( randn(100, 100), 5)) imshow(gaussian_filter( randn(100, 100), 5)) specgram(sin(10000*2*pi *linspace(0,1,44100)), Fs=44100)

32. Brian At last!

36. Online help

37. Anatomy of a Brian script Import the Brian package + Scipy and Pylab Define some parameters, you can use physical units Define neuron equations, the “volt” term says that V has units of volts, and Brian checks the consistency of your equations. Create neurons with given equations and fixed threshold and reset value. Create synapses – here recurrent random connectivity with probability.1 for each pair of neurons and given synaptic weight, spikes cause an instantaneous increase of amount weight in variable V of the target neuron Record spiking activity Initialise state variables to random values. Run simulation Analyse results – do a raster plot

38. Model equations tau=10*ms eqs=''' dv/dt = (v0-v)/tau : volt u=v*v : volt**2 x=u v0 : volt ''' units differential equation equation alias parameter  Non-autonomous equations, use the reserved symbol “t” for time  Stochastic DEs  Use the term “xi” for noise with:  Has units s -1/2. Use typically as “sigma*xi*(2/tau)**0.5”  Solvers:  Exact for linear DEs (detected automatically, method=‘linear’)  Euler for nonlinear DEs (default, method=‘Euler’)  2 nd order Runge-Kutta (default, method=‘RK’)  Exponential Euler (implicit=True, method=‘exponential_Euler’)

39. Neuron groups group=NeuronGroup(N, model=’dv/dt = -v/tau : volt’, threshold=’v>15*mV’, reset=’v=0*mV’, refractory=2*ms, method=’implicit’) model equations threshold condition what happens after spiking refractory period integration method optional units of v group.v is a vector with values of v for all neurons group.v[5] is the value of v for neuron 5 P=group[:100] P=group.subgroup(100) defines a subgroup of 100 neurons Special groups: PoissonGroup(N,rates=10*Hz) SpikeGeneratorGroup(N,spikes) N Poisson processes fires at specific times spikes=[(3,2*ms),(1,10*ms),(2,15*ms)]

40. Connections synapses=Connection(P,Q,’v’) synapses[5,7]=2*mV modified variable P P Q Q synapses.connect_full(P,Q,weight=2*mV) synapses.connect_random(P,Q,sparseness=.02,weight=2*mV) synapses.connect_full(P[50:100],Q[20:40], weight=lambda i,j:w0*exp(-(i-j)**2)) when neuron 5 spikes: Q.v[7]+=2*mV Building connections synapses=Connection(P,Q,’v’,sparseness=.02,weight=2*mV)

41. Delays synapses=Connection(P,Q,’v’,delay=2*ms) Homogeneous delays synapses=Connection(P,Q,’v’,delay=True,max_delay=10*ms) synapses.delay[5,7]=2*ms synapses.connect_full(P,Q,weight=2*mV,delay=(1*ms,2*ms)) synapses.connect_full(P,Q,weight=2*mV, delay=lambda i,j:w0*exp(-(i-j)**2)) Heterogeneous delays

42. Monitors  Recording spike trains  Recording variables M=SpikeMonitor(P) run(100*ms) raster_plot(M) show() M=StateMonitor(P,'v',record=True) run(100*ms) plot(M.times/ms,M[0]/mV) show()

43. Network operations … M=SpikeMonitor(G) … @network_operation def check_too_many_spikes(): if M.nspikes>1000: stop() … run(10*second) this function is called at every time step

44. Examples

45. Integrate-and-fire model  Stimulate an integrate-and-fire model with regularly spaced spikes  Tip: use SpikeGeneratorGroup

46. Integrate-and-fire model

47. Current-frequency curve  Plot the firing rate of an IF model as a function of input current  Tip: counter=SpikeCounter(group) counts spikes for every neuron

48. Current-frequency curve

49. Cortical column  Ring of noisy IF neurons with distance-dependent connections dv/dt=(v0-v)/  +  /   (t) w(i,j)=cos(2  (i-j)/N)w 0

50. Cortical column

51. Reliability  Response of a noisy IF neuron to repeated injections of the same noisy current

52. Reliability

53. Jeffress model décalage temporel δ δ +d left =d right  synchronous inputs  neuron fires

54. Jeffress model delays (sound turning around the head)

56. Interactive session

57.  Try writing your own models with Python, NumPy, SciPy and Brian  We’ll be around to help solve any problems  Take a look at the online documentation too:  http://www.python.org/doc/2.5.4/tut/tut.html http://www.python.org/doc/2.5.4/tut/tut.html  http://www.python.org/doc/2.5.4/ http://www.python.org/doc/2.5.4/  http://docs.scipy.org/doc/ http://docs.scipy.org/doc/  http://matplotlib.sourceforge.net/contents.html http://matplotlib.sourceforge.net/contents.html  http://www.briansimulator.org/docs/ http://www.briansimulator.org/docs/

58. http://www.briansimulator.org Thanks!