Seminar UNICAMP february 2017

1. Phase transitions and self-organized criticality
in networks of stochastic spiking neurons
Osame Kinouchi
Departmento de Física - FFCLRP - USP
IMECC- UNICAMP, Campinas - SP, 22/02/2017

2. Scientific Reports 6: 35831 (2016)
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3. Collaborators at CEPID NEUROMAT
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Ludmila Brochini
Jorge Stolfi
Ariadne A. Costa
Antônio C. Roque Mauro Copellihttp://neuromat.numec.prp.usp.br/team

4. Allan Turing (1950)
Computing machinery and intelligence. Mind, 59, 433-460.
Is there a corresponding phenomenon [criticality]
for minds, and is there one for machines? There
does seem to be one for the human mind. The
majority of them seems to be subcritical, i.e., to
correspond in this analogy to piles of subcritical
size. An idea presented to such a mind will on
average give rise to less than one idea in reply.
A smallish proportion are supercritical. An idea
presented to such a mind may give rise to a whole
"theory" consisting of secondary, tertiary and
more remote ideas. (...) Adhering to this analogy
we ask, "Can a machine be made to be
supercritical?"
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5. Motivation: neuronal avalanches
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6. Distribution of sizes and durations of
neuronal avalanches
Avalanche Size Distribution: PS(s) ∝ s-3/2
Avalanche Duration Distribution: PD(d) ∝ d-2
Power laws with mean-field exponents
Beggs & Plenz, 2003
PS(S)
s = N ∑a
b ρ[t]
d = b − a
 fraction of firing neurons ρ[t]

7. i = 1, 2, …, N neurons
Xi[t] = 1 (spike or firing event) Xi[t] = 0 (silent)
Vi[t+1] = 0 if Xi[t] = 1 (voltage reset)
Vi[t+1] = μVi[t] + I + N-1 ∑j Wij Xj[t] if Xi[t] = 0 (integrate voltage)
Firing event:
Prob( Xi[t+1] = 1 ) = Φ(Vi[t])
Φ = firing probability function
0 ≤ Φ(V) ≤ 1 (stochastic integrate-and-fire neuron)
Our model: Stochastic discrete time spiking
neurons (Galves & Löcherbach, 2013)
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All-to-all network

8. Examples of firing functions
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1
VT VS V
r = 1
r > 1
r < 1
Φ(V) For VT < V < VS: Φ(V) = [ 𝚪(V-VT) ]r
𝚪 = Neuronal Gain
Obs: Integrate-and-fire neuron: VT = VS or 𝚪 → ∞

9. Mean field approach
 Order parameter = fraction of firing neurons: ρ[t] = 1/N ∑i Xi[t]
 Control parameters: W = <Wij> and 𝚪 = < 𝚪i >
 MF approximation: 1/N ∑i Wij Xi[t] = W ρ[t], where W = <Wij>
 Voltages evolve as:
Vi[t+1] = 0 if Xi[t] = 1
Vi[t+1] = μVi[t] + I + Wρ if Xi[t] = 0
9W
ρ
Wc
Phase Transition
from silent state to active state
Critical point with
avalanches

10. Calculating the fraction of firing neurons
(order parameter)
ρ[t] = ∫ Φ(V) pt(V) dV
In the stationary state, the
voltages assume a discrete
stationary set of values Uk.
The density of neurons
with Uk is 𝛈k .
Normalization: ∑ 𝛈k = 1
So, in the stationary state,
ρ = ∑k=1 Φ(Uk) 𝛈k
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pt(V)
V
Transient
Stationary
p(V)
VU2U1 U3U0
𝛈1
𝛈2
𝛈3
Time

11. Stationary distributions
pt(V) dV → P(V) with discrete peaks

12. Mean field stationary states
μ > 0 case (several peaks, only numeric solutions):
Uk = Voltage of the k-th stationary peak, 𝛈k = peak height
ρ = ∑k=1 Φ(Uk) ηk
Uk = μ Uk-1 + I + W ρ
ηk = [1- Φ(Uk-1)] ηk-1
η0 = ρ = 1 - ∑k=1 ηk (Normalization)
12Uk
η0 = ρ η1
η2
η3
η4
Φ
(1- Φ)η0
Φ
Φ
Φ
(1- Φ)η1
(1- Φ)η2
(1- Φ)η3
U1 U2 U3 U4U0 = 0

13. Example: μ > 0, r = 1, VT = 0, I = 0
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14.  Due to the one step refractory period,
we cannot have ρ > 1/2.
 ρ(𝚪b,Wb) = 1/2 defines a bifurcation
line (𝚪b = 2/Wb for the μ = 0 case)
 Cycles-2 occur because there is only
two peaks, with U1 > VS
 Deterministic dynamics since now
Φ(U1 > VS) = 1
𝛈0[t+1] = 𝛈1[t] = 1 - ρ[t]
𝛈1[t+1] = 𝛈0[t] = ρ[t]
 Cycle dynamics:
ρ[t+1] = 1 - ρ[t]
 Solutions occur for any values inside
the region [VS/W, (W-VS)/W]
Observation: marginally stable cycles-2
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VS
Φ(U1) = 1
V
1
VVS
1
Φ(U1) = 1
0
0
𝛈1[t]
𝛈1[t+1]
𝛈1[t+1]
𝛈1[t]
𝛈0[t+1]
𝛈0[t]
𝛈0[t]
𝛈0[t+1]

15. Case μ = 0
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μ = 0 case (only two peaks, analytic solutions):
η0 = ρ = Φ(U1) η1 = Φ(I+Wρ) (1 - ρ)
η1 = 1 - ρ (Normalization)
With:
Φ(x) = 0 for x < VT
Φ(x) = [ 𝚪(x-VT) ]r for VT < x < VS = VT + 1/𝚪
Φ(x) = 1 for x > VS
Very easy Math! (polinomial equations). Solve for 0 < ρ <1/2:
ρ = [𝚪(I+Wρ-VT)]r (1-ρ)
ρ
η1 = 1- ρ
1-Φ
1
Φ
U1

16. Parameters of the firing function Φ
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1
Threshold voltage VT Saturation voltage VS =VT + 1/𝚪 V
r = 1
r > 1
r < 1
Φ(V) For VT < V < VS: Φ(V) = [𝚪(V-VT)]r
𝚪 = Neuronal Gain

17. μ = 0, linear saturating case r = 1, VT = 0
(Larremore et al., PRL 2014) But they do not report phase transitions
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• Case r = 1, solve for 0 < ρ <1/2:
ρ = [𝚪(I + Wρ − VT)] (1 − ρ) or:
𝚪W ρ2 + (1 − 𝚪W + 𝚪I − 𝚪VT) ρ + 𝚪VT − 𝚪I = 0
• Solutions:
• Continuous and discontinuous phase transitions here
Φ
V

18. Case μ = 0, VT = 0, r = 1
Linear saturating model without threshold
 Solutions:
ρ+ = (W - Wc)/ W with Wc = 1/ 𝚪
or
ρ+ = (𝚪 - 𝚪c) / 𝚪 with 𝚪c = 1/ W
ρ− = 0 for W < Wc or 𝚪 < 𝚪 c
Continuous transition with exponent
β = 1 (Directed Percolation
Universality Class)
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ρ = 0
0 < ρ < 1/2
cycles-2

19. Case μ = 0, VT = 0, r = 1, I > 0
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At the critical line 𝚪cWc = 1, for I → 0: ρ ∝ I1/2 χ = dρ/dI ∝ I-1/2
Mean field critical exponent δ = 2
Stevens network psychophysical exponent: m = 1/δ = 1/2
(Kinouchi and Copelli, Nat. Phys. 2006)

20. Case μ = 0, VT = 0, 0 < r < 2
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r = 1
Continuous phase
transition
r = 1.2
Discontinuous phase
transition
r = 2
Transition only to
cycles-2
r < 1
Wc = 0
Φ(V)
𝚪 x W
phase diagram
r=1

21. Case μ = 0, r = 1, threshold VT > 0
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VT = 0 VT = 0.1VT = 0.05

22. Avalanche size distributions in the static
model with Wc =1, 𝚪c = 1
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Complementary
cumulative
distribution function:
CS(s) = ∑
∞
k=s
PS(k) ∝ s-1/2
Linear saturating model: r = 1, VT = 0
PS(s) ∝ s-3/2s = N ∑a
b ρ[t]
d = b − a

23. Avalanche duration distributions in the
static model with Wc =1, 𝚪c = 1
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PD(d) ∝ d-2 Complementary
cumulative
distribution function:
CD(d) = ∑
∞
k=d PD(k) ∝ d-1

24. Self-organization in a continuous phase
transition: dynamic synapses and dynamic gains
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r =1, VT =0
Idea: not dissipation and
loading at the sites but
decreasing and increasing
the links Wij (or the gains 𝚪i)
NEW!
Can dynamical
synapses produce true self-
organized criticality?
Ariadne de Andrade Costa,
Mauro Copelli and Osame
Kinouchi
Journal of Statistical Mechanics:
Theory and Experiment, 2015(6),
P06004

25. Why to separate the average gain 𝚪 from the
average synaptic weight W?
 In a biological network, each
neuron i has a neuronal gain 𝚪i[t]
located at the Axonal Initial
Segment (AIS). Its dynamics is
linked to sodium channels.
 The synapses Wij[t] are located at
the dendrites, very far from the
axon. Its dynamics is due to
neurotransmitter vesicle
depletion.
 So, although in our model they
appear always together as 𝚪W,
this is due to the use of point like
neurons. A neuron with at least
two compartments (dendrite +
soma) would segregate these
variables.
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AIS, 𝚪i[t]
Wij[t]

26. Dynamic synapses vs dynamic gains
 Dynamic synapses (Levina et al., 2007; Levina et al., 2009;
Bonachela et al., 2010; Costa et al., 2015, Campos et al.,
2016):
Wij[t+1] = Wij[t] + 1/𝞽 (A − Wij[t]) − u Wij[t] Xj[t]
 Dynamic gains (Brochini et al., 2016):
𝚪i[t+1] = 𝚪i[t] + 1/𝞽 (A − 𝚪i[t]) − u 𝚪i[t] Xi[t]
 Parameters range:
0 < u < 1, A > Wc (or A > 𝚪c)
Literature with dynamic synapses used
𝞽 = O(N) (non-biological)
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Wc, 𝚪c
A, 𝞽 u
O(N2) equations
O(N) equations!

27. Self-organization of the average gain
toward the critical region
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𝚪* ≃ 𝚪c
𝚪[t] = N-1 ∑i 𝚪i[t]

28. Self-organization slightly above the critical
line by using dynamic neuronal gains
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With u = 1, A = 1.1 𝚪c :
𝚪* ≃ 1.001 𝚪c = self-organized super-criticality (SOSC)

29. For finite 𝞽, we have always self-organized
super-criticality (SOSC)
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𝞽 = 100 ms, u = 1
𝚪* = 1.001
𝞽 = 1000 ms, u = 1
𝚪* = 1.0001
CS = 1

30. Finite 𝞽: self-organized super-criticality
(SOSC) seems to be unavoidable
 But is this bad?
 Perhaps supercritical avalanches exist
but are being filtered away in standard
experiments
 Perhaps SOSC can explain biological
phenomena: large avalanches (dragon
kings), epileptic activity etc.
 Perhaps only large avalanches process
informations and elicit actions (who
notice small avalanches?)
 Perhaps, Turing was right…
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Shaukat & Thivierge, 2016
Front. Comput. Neurosci. 10:29

31. Perpectives: a new neuronal network
formalism to be explored
 More results, perhaps analytic, for the μ > 0 case.
 Better study of dynamic synapses and dynamic gains
 Other specific Φ functions (ex: Φ(V) = [𝚪(V-VT)]r/[1+[𝚪(V-VT)]r] = no 2-
cycles
 Theorems for general Φ functions (ex: all linear piecewise functions give
analytic solutions)
 Self-organized bistability (SOB)
 Other network topologies (scale free, small world etc.)
 Other kinds of couplings
 Inhibitory neurons (ex: Larremore et al., 2014)
 Very large networks: N > 106, synapses > 1010
 Realistic topologies (ex: cortical Potjans-Diesmann model with layers and
different neuron populations, N=8x104, synapses = 3x108, Cordeiro et al.,
2016)
 Etc., etc., etc…
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32. Visit us at Ribeirão Preto!
(we also have research fellowships at Neuromat)
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This paper results from research
activity on the
FAPESP Center for
Neuromathematics (FAPESP grant
2013/07699-0).
OK and AAC also received support
from Núcleo de Apoio à Pesquisa
CNAIPS-USP and FAPESP (grant
2016/00430-3).
LB, JS and ACR also received CNPq
support (grants 165828/2015-3,
310706/2015-7 and 306251/2014-0).