1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
|
#!/usr/bin/env python
"""
Late Emergence of the Whisker Direction Selectivity Map in the Rat Barrel Cortex.
Kremer Y, Leger JF, Goodman DF, Brette R, Bourdieu L (2011).
J Neurosci 31(29):10689-700.
Development of direction maps with pinwheels in the barrel cortex.
Whiskers are deflected with random moving bars.
N.B.: network construction can be long.
"""
from brian2 import *
import time
t1 = time.time()
# PARAMETERS
# Neuron numbers
M4, M23exc, M23inh = 22, 25, 12 # size of each barrel (in neurons)
N4, N23exc, N23inh = M4**2, M23exc**2, M23inh**2 # neurons per barrel
barrelarraysize = 5 # Choose 3 or 4 if memory error
Nbarrels = barrelarraysize**2
# Stimulation
stim_change_time = 5*ms
Fmax = .5/stim_change_time # maximum firing rate in layer 4 (.5 spike / stimulation)
# Neuron parameters
taum, taue, taui = 10*ms, 2*ms, 25*ms
El = -70*mV
Vt, vt_inc, tauvt = -55*mV, 2*mV, 50*ms # adaptive threshold
# STDP
taup, taud = 5*ms, 25*ms
Ap, Ad= .05, -.04
# EPSPs/IPSPs
EPSP, IPSP = 1*mV, -1*mV
EPSC = EPSP * (taue/taum)**(taum/(taue-taum))
IPSC = IPSP * (taui/taum)**(taum/(taui-taum))
Ap, Ad = Ap*EPSC, Ad*EPSC
# Layer 4, models the input stimulus
eqs_layer4 = '''
rate = int(is_active)*clip(cos(direction - selectivity), 0, inf)*Fmax: Hz
is_active = abs((barrel_x + 0.5 - bar_x) * cos(direction) + (barrel_y + 0.5 - bar_y) * sin(direction)) < 0.5: boolean
barrel_x : integer # The x index of the barrel
barrel_y : integer # The y index of the barrel
selectivity : 1
# Stimulus parameters (same for all neurons)
bar_x = cos(direction)*(t - stim_start_time)/(5*ms) + stim_start_x : 1 (shared)
bar_y = sin(direction)*(t - stim_start_time)/(5*ms) + stim_start_y : 1 (shared)
direction : 1 (shared) # direction of the current stimulus
stim_start_time : second (shared) # start time of the current stimulus
stim_start_x : 1 (shared) # start position of the stimulus
stim_start_y : 1 (shared) # start position of the stimulus
'''
layer4 = NeuronGroup(N4*Nbarrels, eqs_layer4, threshold='rand() < rate*dt',
method='euler', name='layer4')
layer4.barrel_x = '(i // N4) % barrelarraysize + 0.5'
layer4.barrel_y = 'i // (barrelarraysize*N4) + 0.5'
layer4.selectivity = '(i%N4)/(1.0*N4)*2*pi' # for each barrel, selectivity between 0 and 2*pi
stimradius = (11+1)*.5
# Chose a new randomly oriented bar every 60ms
runner_code = '''
direction = rand()*2*pi
stim_start_x = barrelarraysize / 2.0 - cos(direction)*stimradius
stim_start_y = barrelarraysize / 2.0 - sin(direction)*stimradius
stim_start_time = t
'''
layer4.run_regularly(runner_code, dt=60*ms, when='start')
# Layer 2/3
# Model: IF with adaptive threshold
eqs_layer23 = '''
dv/dt=(ge+gi+El-v)/taum : volt
dge/dt=-ge/taue : volt
dgi/dt=-gi/taui : volt
dvt/dt=(Vt-vt)/tauvt : volt # adaptation
barrel_idx : integer
x : 1 # in "barrel width" units
y : 1 # in "barrel width" units
'''
layer23 = NeuronGroup(Nbarrels*(N23exc+N23inh), eqs_layer23,
threshold='v>vt', reset='v = El; vt += vt_inc',
refractory=2*ms, method='euler', name='layer23')
layer23.v = El
layer23.vt = Vt
# Subgroups for excitatory and inhibitory neurons in layer 2/3
layer23exc = layer23[:Nbarrels*N23exc]
layer23inh = layer23[Nbarrels*N23exc:]
# Layer 2/3 excitatory
# The units for x and y are the width/height of a single barrel
layer23exc.x = '(i % (barrelarraysize*M23exc)) * (1.0/M23exc)'
layer23exc.y = '(i // (barrelarraysize*M23exc)) * (1.0/M23exc)'
layer23exc.barrel_idx = 'floor(x) + floor(y)*barrelarraysize'
# Layer 2/3 inhibitory
layer23inh.x = 'i % (barrelarraysize*M23inh) * (1.0/M23inh)'
layer23inh.y = 'i // (barrelarraysize*M23inh) * (1.0/M23inh)'
layer23inh.barrel_idx = 'floor(x) + floor(y)*barrelarraysize'
print("Building synapses, please wait...")
# Feedforward connections (plastic)
feedforward = Synapses(layer4, layer23exc,
model='''w:volt
dA_source/dt = -A_source/taup : volt (event-driven)
dA_target/dt = -A_target/taud : volt (event-driven)''',
on_pre='''ge+=w
A_source += Ap
w = clip(w+A_target, 0*volt, EPSC)''',
on_post='''
A_target += Ad
w = clip(w+A_source, 0*volt, EPSC)''',
name='feedforward')
# Connect neurons in the same barrel with 50% probability
feedforward.connect('(barrel_x_pre + barrelarraysize*barrel_y_pre) == barrel_idx_post',
p=0.5)
feedforward.w = EPSC*.5
print('excitatory lateral')
# Excitatory lateral connections
recurrent_exc = Synapses(layer23exc, layer23, model='w:volt', on_pre='ge+=w',
name='recurrent_exc')
recurrent_exc.connect(p='.15*exp(-.5*(((x_pre-x_post)/.4)**2+((y_pre-y_post)/.4)**2))')
recurrent_exc.w['j<Nbarrels*N23exc'] = EPSC*.3 # excitatory->excitatory
recurrent_exc.w['j>=Nbarrels*N23exc'] = EPSC # excitatory->inhibitory
# Inhibitory lateral connections
print('inhibitory lateral')
recurrent_inh = Synapses(layer23inh, layer23exc, on_pre='gi+=IPSC',
name='recurrent_inh')
recurrent_inh.connect(p='exp(-.5*(((x_pre-x_post)/.2)**2+((y_pre-y_post)/.2)**2))')
if get_device().__class__.__name__=='RuntimeDevice':
print('Total number of connections')
print('feedforward: %d' % len(feedforward))
print('recurrent exc: %d' % len(recurrent_exc))
print('recurrent inh: %d' % len(recurrent_inh))
t2 = time.time()
print("Construction time: %.1fs" % (t2 - t1))
run(5*second, report='text')
# Calculate the preferred direction of each cell in layer23 by doing a
# vector average of the selectivity of the projecting layer4 cells, weighted
# by the synaptic weight.
_r = bincount(feedforward.j,
weights=feedforward.w * cos(feedforward.selectivity_pre)/feedforward.N_incoming,
minlength=len(layer23exc))
_i = bincount(feedforward.j,
weights=feedforward.w * sin(feedforward.selectivity_pre)/feedforward.N_incoming,
minlength=len(layer23exc))
selectivity_exc = (arctan2(_r, _i) % (2*pi))*180./pi
scatter(layer23.x[:Nbarrels*N23exc], layer23.y[:Nbarrels*N23exc],
c=selectivity_exc[:Nbarrels*N23exc],
edgecolors='none', marker='s', cmap='hsv')
vlines(np.arange(barrelarraysize), 0, barrelarraysize, 'k')
hlines(np.arange(barrelarraysize), 0, barrelarraysize, 'k')
clim(0, 360)
colorbar()
show()
|
