Supplement to “Learning Rule of Homeostatic Synaptic Scaling: Presynaptic Dependent or Not” by Jian K. Liu, Neural Computation, Vol. 23, No. 12 (December 2011), pp. 3145–3161.
URL: http://www.mitpressjournals.org/doi/abs/10.1162/NECO_a_00210
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Learning Rule of Homeostatic Synaptic Scaling: Presynaptic Dependent or Not: Supplemental Material Jian K. Liu Laboratory of Neurophysics and Physiology, CNRS UMR 8119, Universite Paris Descartes, 75006 Paris, France.
1 Supplementary Color File Color Figure 1, Figure 2, and Figure 3 are shown in the main text.
2 Binary Neural Network Based on the analysis presented in the main text, we conducted numerical simulations to study learning dynamics with a recurrent neural network of binary neurons.
2.1 Binary Neuron We considered the McCulloch & Pitts binary neuron model (?) with discrete-time dynamics, where a neuron is a binary threshold unit. The state of neuron i at time t within (τ )
the trial τ is denoted as si (t) ∈ [0, 1], where 1 or 0 represents as firing or not firing.
Then the neural dynamics has the following form ) ( ∑ (τ ) (τ ) (τ ) si (t + 1) = H wij sj (t) + Ii (t) − Θ ,
t = 1, . . . , tmax ,
(1)
j (τ )
where wij > 0 denotes excitatory synaptic weight from presynaptic neuron j to postsynaptic neuron i at trial τ . tmax is the maximal running time of one trial. The constant Θ = 1 acts as the firing threshold. Ii (t) = 1 is the stimulus for neuron i. H(x) is the Heaviside function such that there is a spike whenever the total current crosses the threshold, otherwise no spike occurs. At the end of every trial, neuron states are re(τ )
(0)
set as their initial conditions as si (0) = si (0), ∀τ . Another setting can also be used: (τ +1)
si
(τ )
(0) = si (tmax ), which gives a continuous neural dynamics over trials. Two reset
schemes have similar behaviors since network dynamics are discrete without decaying dynamics and persistent activity at later time t.
2.2
Simulation Parameters
In all simulations, the network includes 100 excitatory neurons with a connection probability 25% between any pair of neurons chosen randomly. The stimulus is a brief impulse to the network such that 4 excitatory neurons fire at t = 2, which is analogous to a synchronous input composed by a small subset of neurons. The synaptic connection topology is fixed, i.e., synaptic weights never change their signs, and upper and lower bounds of weights are introduced as wmax = 0.6 and wmin = wmax /1000. Thus, weights are always positive, and synapse never die to zero. Self-connections are excluded. wmax = 0.6 is a value that requires at least 2 synchronous presynaptic inputs to fire a postsynaptic cell. The learning parameters are αν¯ = 0.2 and αw = 0.01. The target firing rate νgoal = 1 for all neurons to make network display a sparsely spatiotemporal activity pattern. Results are robust to variations of parameters, such as the network size, the connection probability, weight bounds. A ready-to-use MATLAB code is available online at the author’s homepage.
2.3 Results of Simulations Results are presented in Figure 4, Figure 5, and Figure 6, which are comparable with Figure 1, Figure 2, and Figure 3 from spiking neuron network in the main text. 2
Cell
Cell
A
B 100 200 300 400 500
τ=1
τ=169
τ=1
τ=200
100 200 300 400 500
τ=170
τ=173
τ=300
τ=500
C
100 time
200
100 time
200
100 200 time (ms)
mean ν¯
(τ )
4
ν¯(τ +1) − ν¯(τ )
D
2 0 0 1
100 200 time (ms) 1
SS PSD
100
0.5
200
300
400
500
0 −1 0
0 600 0.02 0
100
200
300 τ (trial)
400
500
−0.02 600
Figure 1: Network dynamics are unstable under SS and stable under PSD. (A) Raster patterns under SS at τ = 1, 169, 170 and 173. (B) Raster patterns under PSD at τ = 1, 200, 300 and 500. White dots are spikes. (C) Mean firing rate ν¯(τ ) averaged over all neurons exhibits large oscillations under SS, and stably converges to the target under PSD. (D) ν¯(τ +1) − ν¯(τ ) indicates the degree of jump discontinuity. Excitation explosion is exhibited under SS but depressed under PSD. In (A-B) all indexes of neurons (y−axis) are sorted according to their spiking time after learning. Data under SS are colored with blue, and red under PSD in all figures.
3
Pre-strength swi
A
Post-strength swj
C
30
B SS
τ=1 τ=100 τ=300
20
10
PSD
τ=1 τ=300 τ=800
5
10 0 0 30
100
200 300 Post-cell j
0 0
400
D
SS
30
20
20
10
10
0 0
100
4
PSDpre
200 300 Pre-cell i
E
400
0 0
100
200 300 Post-cell j
400
100
200 300 Pre-cell i
400
PSD
σsw
SSpre SSpost
2 0 0
PSDpost
100
200
300 τ (trial)
400
500
600
Figure 2: Synaptic competition is realized by PSD, not SS. (A) Pre-strengths swi under SS are distributed uniformly within one trial and scaled globally across different trials τ = 1, 100 and 300. (B) swi under PSD are distributed and changed heterogeneously, particularly at τ = 300. (C) Post-strengths swj under SS are scaled globally across trials, even they are distributed less uniformly within one trial. (D) swj under PSD are heterogeneous both within one trial and across trials. (E) Standard deviations σsw of SSpre in (A), PSDpre in (B), SSpost in (C) and PSDpost in (D), are significantly different and separated under PSD and nearly overlapping under SS, which indicates that synaptic competition is missed under SS but exhibited under PSD.
4
B
10
Ratios
0.03 r1
120
˜ (τ ) ) ρ(D
1
˜ (τ ) ) σ1 (T
A
r2
0 0
200
400
0 600
C
0 0
200
400
0 600
D ρ(D(τ ) ) 0 0
200 400 τ (trial)
0 600
1.01
48.6
1 0
200 400 τ (trial)
σ1 (T(τ ) )
4
ρ(W(τ ))
15
48.4 600
Figure 3: Synaptic matrix is convergent under PSD, not SS. (A) r1 (solid line, blue) and r2 (dash line, blue) are close to the theoretical bound 1 under SS. r1 (solid line, red) and r2 (dash line, red) are 100-fold less under PSD. (B) The spectral norm of allstep transition matrix, (C) the largest eigenvalue of synaptic matrix and (D) the spectral norm of one-step transition matrix are always larger than 1, and convergent under PSD, but not convergent under SS.
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A Cell
20 40 60 80 100
τ=1
τ=208
τ=1
τ=200
Cell
B
20 40 60 80 100
τ=209
τ=211
τ=300
τ=800
C
10 time
20
10 time
20
10 time
20
10 time
mean ν¯(τ )
10
ν¯(τ +1) − ν¯(τ )
D
5 0 0 5
1 SS PSD
0.5
200
400
600
800
0 −5 0
20
0 1000 0.02 0
200
400 τ (trial) 600
800
−0.02 1000
Figure 4: (A) SS develops unstable network dynamics with excitation explosion, where raster patterns at τ = 1, 208, 209, 211 are displayed. White dots are spikes. (B) PSD produces a stable network dynamics developed at different learning phases τ = 1, 200, 300, 800. (C) Mean firing rate ν¯(τ ) averaged over all neurons exhibits large oscillation under SS, but stably converges to the target under PSD. (D) Excitation explosion is represented by a jump discontinuity under SS indicated by ν¯(τ +1) − ν¯(τ ) . Stability is maintained during learning without discontinuity under PSD. Data of SS are colored with blue, and PSD with red.
6
Pre-strength swi
A
Post-strength swj
C
20
B SS
τ=1 τ=100 τ=200
10
6
PSD
τ=1 τ=300 τ=800
4 2
0 0 15
50 Post-cell j
0 0
100
D
SS
15
10
10
5
5
0 0
50 Pre-cell i
E 4
100
50 Pre-cell i
100
PSD
0 0
100
50 Post-cell j
SSpre
σsw
PSDpre
2 0 0
SSpost PSDpost
200
400
600
800
1000
τ (trial)
Figure 5: (A) Pre-strength swi of all neurons in the network under SS changes uniformly at τ = 1, 100, 200. (B) swi under PSD changes heterogeneously at τ = 1, 300, 800. (C) Post-strength swj under SS is homogeneous. (D) swj under PSD is heterogeneous. (E) Standard deviation σsw of per- and post-strength, SSpre in (A), PSDpre in (B), SSpost in (C) and PSDpost in (D), are different under PSD and similar under SS. The first 4 neurons are stimulated.
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B
15
Ratios
0.01 r1
1000
˜ (τ ) ) ρ(D
1
˜ (τ ) ) σ1 (T
A
r2
0 0
500
0 1000
C
0 0
500
0 1000
D
500 τ (trial)
0 1000
1.01
σ1 (T(τ ) )
ρ(D(τ ) ) 0 0
100.3
2
ρ(W(τ ))
10
1 0
500 τ (trial)
99.9 1000
Figure 6: (A) r1 (solid line) and r2 (dash line) are close to 1 under SS (blue), and smaller under PSD (red). (B) Spectral norm of transition matrix is increasing under SS and convergent under PSD. The maximal eigenvalue of synaptic matrix (C) and the spectral norm of one step transition matrix (D) are always larger than 1. All curves of PSD are convergent. Data of SS are colored with blue, and PSD with red.
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