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Neural modeling and simulation
Romain Brette
Ecole Normale Supérieure
with
http://briansimulator.org
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
A neural network with
Pi Pe
Ce
CiP
from brian import *
eqs='''dv/dt = (ge+gi-(v+49*mV))/(20*ms) : voltdge/dt = -ge/(5*ms) : voltdgi/dt = -gi/(10*ms) : volt''‘
P=NeuronGroup(4000,model=eqs, threshold=-50*mV,reset=-60*mV)
P.v=-60*mV+10*mV*rand(len(P))Pe=P.subgroup(3200)Pi=P.subgroup(800)
Ce=Connection(Pe,P,'ge',weight=1.62*mV,sparseness=0.02)Ci=Connection(Pi,P,'gi',weight=-9*mV,sparseness=0.02)
M=SpikeMonitor(P)
run(1*second)
raster_plot(M)show()
Part I –Neurons
Equivalent electrical circuit
I
Linear approximation of leak current: I = gL(Vm-EL)
leak or resting potential
leak conductance = 1/R membrane resistance
= capacitance
EL -70 mV : the membrane is « polarized » (Vin < Vout)
The membrane equationIinj
Iinj
outside
inside
injLmm I
R
EV
dt
dVC
)(
=1/R
Vm
injmLm RIVE
dt
dV
RC membrane time constant(typically 3-100 ms)
tau = 10*msR = 50*MohmEL = -70*mVIinj = 0.5*nA
eqs='''dvm/dt = (EL-vm+R*Iinj)/tau : volt''‘
The integrate-and-fire model
spike threshold
action potential
PSP
« Integrate-and-fire »: RIVEdt
dVmL
m
If V = Vt (threshold)then: neuron spikes and V→Vr (reset)
(phenomenological description of action potentials)
« postsynaptic potential »
Current-frequency relationship1
log1
tL
rL
VRIE
VRIE
TF
from brian import *
N = 1000tau = 10 * mseqs = '''dv/dt=(v0-v)/tau : voltv0 : volt'''group = NeuronGroup(N, model=eqs, threshold=10 * mV, reset=0 * mV)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()
Refractory period
Δ = refractory period
tL
rL
VRIE
VRIET
log from reset
1
log1
tL
rL
VRIE
VRIE
TF max 1/Δ
from 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()
Synaptic currents
synaptic current
postsynaptic neuron
synapse
Is(t)
smLm RIVE
dt
dV
Idealized synapse Total charge Opens for a short duration Is(t)=Qδ(t)
sIQ
Dirac function
)(tRQVEdt
dVmL
m EL
t
Lm eRQ
EtV
)(
RQVV
VEdt
dV
mm
mLm
Spike-based notation:
at t=0=w « synaptic weight »
Example: 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()
A more realistic synapse model Electrodiffusion: )( msss VEgI
ionic channel conductance
synaptic reversal potential
gs(t)
presynaptic spike
openopen closedclosed
))(( mssmLm VEtRgVE
dt
dV
« conductance-based integrate-and-fire model »
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
Example of kinetic model
Post-synaptic effect:
ss
s
mssmLLm
gdt
dg
VEgVEgdt
dVC
)()(
τs=1/βIncoming spike: ss gg
Example: random network
taum = 20 * mstaue = 5 * mstaui = 10 * msEe = 0 * mVEi = -80 * mVEl = -60 * mV
eqs = '''dv/dt = (El-v+ge*(Ee-v)+gi*(Ei-v))/taum : voltdge/dt = -ge/taue : 1dgi/dt = -gi/taui : 1 ''‘
P = NeuronGroup(4000, model=eqs, threshold=10 * mvolt, \ reset=-60 * mvolt, refractory=5 * msecond)Pe = P.subgroup(3200)Pi = P.subgroup(800)we = 6. / 10. # excitatory synaptic weight (voltage)wi = 67. / 10. # inhibitory synaptic weightCe = 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
run(1 * second)
Pi Pe
Ce
CiP
(conductances in units of the leak conductance)
Linearization
Linear approximation:
)()( mii
imLLm VEgVEg
dt
dVC
non-linear
Limi EEVE 2
LTimi
EVEVE
ou
VT ≈ -50 mV
GABA-B(-100 mV)
GABA-A (-70 mV)
AMPA/NMDA(0 mV)
EL ≈ -70 mV
good
bad
ok
Example: random network
Pi Pe
Ce
CiP
from brian import *
eqs='''dv/dt = (ge+gi-(v+49*mV))/(20*ms) : voltdge/dt = -ge/(5*ms) : voltdgi/dt = -gi/(10*ms) : volt''‘
P=NeuronGroup(4000,model=eqs, threshold=-50*mV,reset=-60*mV)
P.v=-60*mV+10*mV*rand(len(P))Pe=P.subgroup(3200)Pi=P.subgroup(800)
Ce=Connection(Pe,P,'ge',weight=1.62*mV,sparseness=0.02)Ci=Connection(Pi,P,'gi',weight=-9*mV,sparseness=0.02)
M=SpikeMonitor(P)
run(1*second)
raster_plot(M)show()
currents
The postsynaptic potential Postsynaptic potential (PSP) = response to a
presynaptic spike for variable Vm(t).
)( Lii
imLm EEgVg
dt
dVC
),...,,,(
),...,,,(
21
21
niiii
ji
ji
niiiii
i
xxxgfdt
dx
xxxgfdt
dg
Spike at time t=0:
ini
ni wxx
Vm(t) = PPSi(t)synaptic variables
Temporal and spatial integration Response to a set of spikes {ti
j} ?
Linearity:
i = synapsej = spike number
ji
jiim ttPPStV
,
)()( Superposition principle
If the differential system is linear:
(example: the « spike response model »)
From integral to differential representation Experimental recordings = integral representation
model?
))/exp()/(exp()( 21 ttatVm
parametric estimation
biexponental
xdt
dx
Vxdt
dVm
m
2
1
(tool: Laplace transform)
xx
21
2
Voltage-gated channels: biophysics of spike initiation
depolarization(Vm ↑)
Na+ Cl-
K+
channels open:Na+ enters
channels inactivate:no current
Rest: Na+ channels are closed
repolarization(Vm ↓)
The sodium channels
Na+
Cl-K+
heterogeneous distribution of charges -> protein conformation can change with potential
Sodium enters when the « gate » is open
Two stable conformations:open and closed
State transitions
Na+
Cl-K+
closed → open
transition requires energy proportional to V
transition rate prop. to T
aV
e
(transition probability in [t,t+dt]
prop. to )dte T
aV
id. open → closed
transition rate prop. to T
bV
e
and ab<0
State transitions
C ⇄ Oα(V)β(V)
Macroscopic equation (many channels):
mVmVdt
dm)()1)((
opening rate
closing rate
m = proportion of open channels
Kinetic equation
mVmdt
dmVm )()(
mVmVdt
dm)()1)((
)()(
1)(
VVVm
time constant
)()(
)()(
VV
VVm
equilibrium value
k
VVVm
21
exp1
1)(
sigmoidal
The sodium current
mVmdt
dmVm )()(
)( VEmgI Na
max. conductance (= all channels open)
reversal potential(= 50 mV)
mVmdt
dmV
VEmgVEgdt
dVC
m
Nall
)()(
)()(
The Hodgkin-Huxley model
Model of the squid giant axon
Nobel Prize 1963
nVndt
dnV
hVhdt
dhV
mVmdt
dmV
VEngVEhmgVEgdt
dVC
n
h
m
KKNall
)()(
)()(
)()(
)()()( 43
the sodium channel has 3 independent « gates »
4 gates
The Hodgkin-Huxley model
Other voltage-dependent channels Other channels open depending on potential.
mVmdt
dmV
VEmgI
mmm
m
)()(
)(
max conductanceproportion of open channels
time constant
equilibrium value
Na+ (sodium)
K+ (potassium) – many different types
Ca2+ (calcium)
many other types of channels
Example: random network
eqs = '''dv/dt = (gl*(El-v)+ge*(Ee-v)+gi*(Ei-v)-\ g_na*(m*m*m)*h*(v-ENa)-\ g_kd*(n*n*n*n)*(v-EK))/Cm : volt dm/dt = alpham*(1-m)-betam*m : 1dn/dt = alphan*(1-n)-betan*n : 1dh/dt = alphah*(1-h)-betah*h : 1dge/dt = -ge/taue : siemensdgi/dt = -gi/taui : siemensalpham = 0.32*(mV**-1)*(13*mV-v+VT)/ \ (exp((13*mV-v+VT)/(4*mV))-1.)/ms : Hzbetam = 0.28*(mV**-1)*(v-VT-40*mV)/ \ (exp((v-VT-40*mV)/(5*mV))-1)/ms : Hzalphah = 0.128*exp((17*mV-v+VT)/(18*mV))/ms : Hzbetah = 4./(1+exp((40*mV-v+VT)/(5*mV)))/ms : Hzalphan = 0.032*(mV**-1)*(15*mV-v+VT)/ \ (exp((15*mV-v+VT)/(5*mV))-1.)/ms : Hzbetan = .5*exp((10*mV-v+VT)/(40*mV))/ms : Hz'''
P = NeuronGroup(4000, model=eqs, threshold=EmpiricalThreshold(threshold= -20 * mV, refractory=3 * ms), implicit=True)trace = StateMonitor(P, 'v', record=[1, 10, 100])
Adaptation
from brian import *
PG = PoissonGroup(1, 500 * Hz)eqs = '''dv/dt = (-w-v)/(10*ms) : voltdw/dt = -w/(30*ms) : volt # the adaptation current'''# The adaptation variable increases with each spikeIF = NeuronGroup(1, model=eqs, threshold=20 * mV, reset='''v = 0*mV w += 3*mV ''')
C = Connection(PG, IF, 'v', weight=3 * mV)
MS = SpikeMonitor(PG, True)Mv = StateMonitor(IF, 'v', record=True)Mw = StateMonitor(IF, 'w', record=True)
run(100 * ms)
plot(Mv.times / ms, Mv[0] / mV)plot(Mw.times / ms, Mw[0] / mV)
show()
membrane potential
adaptation current
linearized K+ current
Threshold adaptationfrom brian import *
eqs = '''dv/dt = -v/(10*ms) : voltdvt/dt = (10*mV-vt)/(15*ms) : volt'''
reset = '''v=0*mVvt+=3*mV'''
IF = NeuronGroup(1, model=eqs, reset=reset, threshold='v>vt')IF.rest()PG = PoissonGroup(1, 500 * Hz)
C = Connection(PG, IF, 'v', weight=3 * mV)
Mv = StateMonitor(IF, 'v', record=True)Mvt = StateMonitor(IF, 'vt', record=True)
run(100 * ms)
plot(Mv.times / ms, Mv[0] / mV)plot(Mvt.times / ms, Mvt[0] / mV)
show()
cortical neuron in vivo (V1)
By the way: the IF model is not a bad model of cortical neurons
Injected current (slice)
Fast spiking cortical cell
IF model with adaptive threshold
(from INCF competition)
The precision of spike timing
Mainen & Sejnowski (1995)
The same constant current is injected 25 times.
The timing of the first spike is reproducible.
The timing of the 10th spike is not.
In cortical neurons in vitro
With IF neuronsfrom brian import *
N = 25tau = 20 * mssigma = .015eqs_neurons = '''dx/dt=(1.1-x)/tau+sigma*(2./tau)**.5*xi:1'''neurons = NeuronGroup(N, model=eqs_neurons, threshold=1, reset=0, refractory=5 * ms)spikes = SpikeMonitor(neurons)
run(500 * ms)raster_plot(spikes)show()
gaussian white noise
With fluctuating current
Mainen & Sejnowski (1995)
The same temporally variable current is injected 25 times.
Spike timing is reproductible even after 1 s.
(cortical neuron in vitro, somatic injection)
IF neurons and fluctuating currenttau_input = 5 * msinput = NeuronGroup(1, model='dx/dt=-x/tau_input+(2./tau_input)**.5*xi:1')
tau = 10 * mssigma = .015eqs_neurons = '''dx/dt=(0.9+.5*I-x)/tau+sigma*(2./tau)**.5*xi:1I : 1'''neurons = NeuronGroup(25, model=eqs_neurons, threshold=1, reset=0, refractory=5 * ms)neurons.I = linked_var(input,'x‘)spikes = SpikeMonitor(neurons)
run(500 * ms)raster_plot(spikes)show()
Part II – Networks
A few examples
Localization of preys by scorpions
Inhibition of opposite neuron
→ more spikes on the source side
(polar representation of firing rates)Conversion temporal code → rate code
Code
Sturzl, W., R. Kempter, and J. L. van Hemmen (2000). Theory of arachnid prey localization. Physical Review Letters 84 (24), 5668
Sound localization by coincidence detection:Jeffress model
delay δδ+dleft=dright
synchronous inputs the neuron fires
Codedefaultclock.dt = .02 * mssound = TimedArray(10 * randn(50000)) # white noise
max_delay = 20 * cm / (300 * metre / second)angular_speed = 2 * pi * radian / second # 1 turn/secondtau_ear = 1 * mssigma_ear = .1eqs_ears = '''dx/dt=(sound(t-delay)-x)/tau_ear+sigma_ear*(2./tau_ear)**.5*xi : 1delay=distance*sin(theta) : seconddistance : second # distance to the centre of the head in time unitsdtheta/dt=angular_speed : radian'''ears = NeuronGroup(2, model=eqs_ears, threshold=1, reset=0, refractory=2.5 * ms)ears.distance = [-.5 * max_delay, .5 * max_delay]traces = StateMonitor(ears, 'x', record=True)
N = 300tau = 1 * mssigma = .1eqs_neurons = '''dv/dt=-v/tau+sigma*(2./tau)**.5*xi : 1'''neurons = NeuronGroup(N, model=eqs_neurons, threshold=1, reset=0)synapses = Connection(ears, neurons, 'v', structure='dense', delay=True, max_delay=1.1 * max_delay)synapses.connect_full(ears, neurons, weight=.5)synapses.delay[0, :] = linspace(0 * ms, 1.1 * max_delay, N)synapses.delay[1, :] = linspace(0 * ms, 1.1 * max_delay, N)[::-1]spikes = SpikeMonitor(neurons)
Sound
Receptors at the two ears
Coincidence detectors
Delay lines
Simulation of Jeffress model
dela
ys
(sound turning around the head)
« Synfire chains »: propagation of synchronous activity
(Die
smann e
t al, 1
99
9)
Layers of excitatory neurons:
each neuron = integrate-and-fire + noise
Neurons in layer 1 are simultaneously activated: propagation
If fewer neurons are activated, no propagation
(« Synfire chains »: term introduced par Abeles)
Synfire chains
layer 1 layer 2
standard deviation
a = number of spikes
Trajectories in space (,a)
attractor
synchronous propagation
dissipation
CodeVr = -70 * mVVt = -55 * mVtaum = 10 * mstaupsp = 0.325 * msweight = 4.86 * mVeqs = '''dV/dt=(-(V-Vr)+x)*(1./taum) : voltdx/dt=(-x+y)*(1./taupsp) : voltdy/dt=-y*(1./taupsp)+25.27*mV/ms+\ (39.24*mV/ms**0.5)*xi : volt'''# Neuron groupsP = NeuronGroup(N=1000, model=eqs, threshold=Vt, reset=Vr, refractory=1 * ms)Pinput = PulsePacket(t=50 * ms, n=85, sigma=1 * ms)# The network structurePgp = [ P.subgroup(100) for i in range(10)]C = Connection(P, P, 'y')for i in range(9): C.connect_full(Pgp[i], Pgp[i + 1], weight)Cinput = Connection(Pinput, Pgp[0], 'y')Cinput.connect_full(weight=weight)# Record the spikesMgp = [SpikeMonitor(p) for p in Pgp]Minput = SpikeMonitor(Pinput)monitors = [Minput] + Mgp# Setup the network, and run itP.V = Vr + rand(len(P)) * (Vt - Vr)run(100 * ms)
Spontaneous activity in a ring
tau = 10 * msN = 100v0 = 5 * mVsigma = 4 * mV
group = 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()
Part III - Plasticity
Hebb’s rule
D. Hebb
When an axon of cell A is near enough to excite cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing B, is increased. (1949)
A B
Neuron A and neuron B are active: wAB increases
Physiologically: « synaptic plasticity »
PSPPSP size is increased
(or: transmission probability is increased)
Synaptic plasticity at spike level
Presynaptic
spike
Post
syn
apti
c sp
ike
Dan &
Poo (
20
06
)
pre post: potentiation post pre: depression
• causal rule• favors synchronous inputs
(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
0s if )(
0s if )(
)(
/
/
,
post
pre
spost
spre
ji
ipre
ipost
eAsf
eAsf
ttfw
Synaptic plasticity with Briansynapses=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
synapses=Connection(input,neurons,'ge') stdp=ExponentialSTDP(synapses,tau_pre,tau_post,dA_pre,dA_post, wmax=gmax)
relative to wmax:
prepre dAwA max
Complete code
Song, S., K. D. Miller, and L. F. Abbott (2000). Competitive hebbian learning through spike timing-dependent synaptic plasticity. Nature Neurosci 3, 919-26.
A few properties of STDP Stability if:
Stationary distribution is bimodal
0 postpostprepre AA (depression > potentiation)
competitive mechanism
(inputs = Poisson spike trains)
N.B.: not bimodal if weight modification is multiplicative
Properties (2):
Stationary synaptic weights
not correlated correlated
Correlated inputs are favored
Properties (3): After convergence, firing is irregular (balanced
regime)
Short-term synaptic plasticity Synaptic efficacy depends on recent activity
depression(typically: exc exc)
facilitation(typically : exc inh)
Phenomenological model
uUdt
du
xdt
dx
f
d
1
Presynaptic spike:
)1(
)1(
uUuu
uxx
Synaptic efficacy: 1,0ux
depression
facilitation 1,0U
decreases by u*x (resource consumption)
increases (sensitization)
synapses=Connection(input,neurons,'ge') stp=STP(synapses,taud=50*ms,tauf=1*ms,U=0.6)
With Brian:
x = synaptic « resources »u = proportion of resources consumed by a spike
Example: facilitationtau_e = 3 * mstaum = 10 * msA_SE = 250 * pARm = 100 * MohmN = 10
eqs = '''dx/dt=rate : 1rate : Hz'''
input = NeuronGroup(N, model=eqs, threshold=1., reset=0)input.rate = linspace(5 * Hz, 30 * Hz, N)
eqs_neuron = '''dv/dt=(Rm*i-v)/taum:voltdi/dt=-i/tau_e:amp'''neuron = NeuronGroup(N, model=eqs_neuron)
C = Connection(input, neuron, 'i')C.connect_one_to_one(weight=A_SE)stp = STP(C, taud=1 * ms, tauf=100 * ms, U=.1trace = StateMonitor(neuron, 'v', record=[0, N - 1])
run(1000 * ms)subplot(211)plot(trace.times / ms, trace[0] / mV)subplot(212)plot(trace.times / ms, trace[N - 1] / mV)show()
regular spike trains
Python in 15 minutes
see also: http://docs.python.org/tutorial/
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
Writing a script If you use IDLE, click on File>New window.
Otherwise use any text editor.
Press F5 to execute
A simple program
# Factorial functiondef factorial(x): if x == 0: return 1 else: return x * factorial(x-1)
print factorial(5)
commentfunction definitionuntyped argument
condition
function calldisplay
structure by indentation(block = aligned instructions)
Numerical objects Base types: int, long, float, complex
Other numerical types (vectors, matrices) defined in the Numpy library (in a few minutes)
x=3+2x+=1y=100Lz=x*(1+2j)u=2.3/7x,y,z = 1,2,3a = b = 123
Control structures
x = 12if x < 5 or (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
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+nameprint 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…
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
Stringsa="Bonjour"b='hello'c="""Une phrasequi n'en finit pas"""
print a+bprint a[3:7]
print b.capitalize()
multiline string
≈ list of characters
many methods (find, replace, split…)
Dictionariesdico={‘one':1,‘two':2,‘three':‘several'}
print dico[‘three']
dico[‘four']=‘many‘del dico[‘one']
key value
for key in dico: print key,'->',dico[key]
iterate all keys
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
Modulesimport mathprint math.exp(1)
from math import expprint 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)
Scipy & Pylab
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*xy=dot(M,x)
from scipy.optimize import *print fsolve(lambda x:(x-1)*(x-3),2)
vector
matrix
matrix product
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)
matrix1 2 34 5 6
matrix1 11 11 1
matrix product
Operationsx+yx-yx*yx/yx**2exp(x)sqrt(x)
dot(x,y)dot(M,x)
M.TM.max()M.sum()
size(x)M.shape
element-wise
dot productmatrix product
transpose
x²
total number of elements
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 columnelements x[1] and x[3]
x[1]=0, x[3]=1add vector x to the 2nd row of M
M[i,:] is a « view » on matrix M copy ( reference)
y=M[0,:]y[2]=5
x=zx[1]=3
M[0,2] is 5
z[1] is 3 x=z.copy()copy:
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 1szero matrixidentity matrixdiagonal matrix
random vector in (0,1)random gaussian vector
0,1,2,...,9
100 numbers between 0 and 1
SciPy example: optimisation
Simple example, least squares polynomial fit
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
Pylab
Pylab 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
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)
Brian
At last!
Online help
Books Theoretical neuroscience, by Dayan &
Abbott, MIT Press
Spiking neuron models, by Gerstner & Kistler, Cambridge Univerity Press
Dynamical systems in neuroscience, by Izhikevich, MIT Press
Biophysics of computation, by Koch, Oxford University Press
Introduction to Theoretical Neurobiology, by Tuckwell, Cambridge University Press
