Talk in Brain Criticality Workshop 2020 (NIH) Dietmar Plenz (org.)

1. H O M E O S T A T I C
C R I T I C A L I T Y I N
S T O C H A S T I C I N T E G R A T E -
A N D - F I R E N E U R O N S
O S A M E K I N O U C H I
P H Y S I C S D E P A R T M E N T - F F C L R P
S Ã O P A U L O U N I V E R S I T Y - B R A Z I L
B R A I N C R I T I C A L I T Y V I R T U A L W O R K S H O P
W E D N E S D A Y O C T O B E R 7 , 2 0 2 0

2. H O M E O S T A T I C C R I T I C A L I T Y =
S E L F - O R G A N I Z E D Q U A S I - C R I T I C A L I T Y
( S O Q C ) W I T H O U T I N F I N I T E S E P A R A T I O N O F
T I M E S C A L E S
Three main homeostatic
mechanisms:
• Homeostatic Synapses 𝑊𝑖𝑗 𝑡 :
𝑁𝐾 synaptic equations
• Homeostatic Neuronal Gains
Γ𝑖[𝑡]:
𝑁 gain equations
• Homeostatic FiringThresholds

3. H O M E O S TAT I C S Y N A P S E S
M E C H A N I S M S O F S E L F - O R G A N I Z E D
Q U A S I - C R I T I C A L I T Y I N N E U R O N A L N E T W O R K S M O D E L S
• Markran-Tsodyks (MT) model:

4. H O M E O S TAT I C S Y N A P S E S
• Levina-Hermann-Geisel (LHG) model (constant u):

5. D I S C R E T E T I M E ( L G H ) S Y N A P S E S

6. D I S C R E T E T I M E S T O C H A S T I C
I N T E G R AT E - A N D - F I R E N E U R O N
• 𝑠𝑖 ∈ 0,1 𝑖 = 1, ⋯ , 𝑁
• 𝑉𝑖 𝑡 + 1 = 𝜇𝑉𝑖 𝑡 + 𝐼𝑖 𝑡 + 𝑗
𝑁
𝑊𝑖𝑗 𝑡 𝑠𝑗[𝑡] if 𝑠𝑖 𝑡 = 0
• 𝑉𝑖 𝑡 + 1 = 0 if 𝑠𝑖 𝑡 = 1
• 𝑃 𝑠𝑖 = 1 = Φ 𝑉𝑖
• 0 < 𝜇 < 1 = leakage parameter
• 𝐼𝑖 = external current
• 𝑊𝑖𝑗 = synaptic weight

7. F I R I N G F U N C T I O N S ɸ ( V )
• Linear-saturating Φ 𝑉 :
• Φ 𝑉𝑖 = 0 if 𝑉𝑖 ≤ 𝜃𝑖
• Φ 𝑉𝑖 = Γ 𝑉𝑖 − 𝜃𝑖 if 𝜃𝑖 < 𝑉𝑖 < 𝑉𝑖
𝑠
= 𝜃𝑖 + 1/Γ
• Φ 𝑉𝑖 = 1 if 𝑉𝑖 ≥ 𝑉𝑖
𝑠
• Rational Φ 𝑉𝑖 :
• Φ 𝑉𝑖 =
Γ 𝑉𝑖−𝜃𝑖
1+Γ 𝑉𝑖−𝜃𝑖
Θ 𝑉𝑖 − 𝜃𝑖
θ VS
1
ɸ(V)
θ
ɸ(V)
1
𝚪
𝚪

8. M T A N D L H G D Y N A M I C S A R E
S U F F I C I E N T B U T N O T N E C E S S A R Y
M E C H A N I S M S
which has a form very similar to sandpile models if 1/𝛕 → 0, u → 0, u/𝛕 → 0

9. P R O P O S E D I N 1 9 9 8 B U T
U N F O R T U N AT E LY N O T S T U D I E D . . .

10. Only N equations,
notice si , not sj

11. Stochastic oscillations and avalanches with homeostatic
neuronal gain

12. 𝑌 =
𝐼
𝜃
𝑔 =
𝑊𝐼𝐼
𝑊𝐸𝐸
=
𝑊𝐼𝐼
𝐽

15. H O M E O S T A T I C S E T P O I N T : 𝑾 ∗
, 𝒉 ∗
= 𝑰 − ( 𝟏 − 𝝁 ) 𝜽 ∗
15
In the 𝑊, ℎ coordinates:
𝑊∗
= 𝑊
𝑐 +
𝐴
1 + 𝜏𝑊𝑢𝑊
ℎ∗
=
1
𝑐 𝜏𝜃𝑢𝜃
2
In the 𝑔, 𝑌 coordinates:
𝑔∗ =
𝑔𝑐
1 + 1/(𝜏𝑊𝑢𝑊)
+
𝑝 − 𝐴/𝐽
𝑞(1 + 𝜏𝑊𝑢𝑊)
𝑌∗
= 𝑌𝑐 1 −
1
𝑐𝐼𝜏𝜃𝑢𝜃
2
For large separation of time scales (𝜏𝑊, 𝜏𝜃 > 100 ms):
𝑊∗
, ℎ∗
, 𝑔∗
, 𝑌∗
→ 𝑊
𝑐 , ℎ𝑐 , 𝑔𝑐 , 𝑌𝑐

16. F I X E D P O I N T I S A F O C U S ( W H I T E B U L L E T ) C L O S E T O
T H E C R I T I C A L P O I N T ( R E D B U L L E T ) .
F I N I T E S I Z E N O I S E ( D E M O G R A P H I C N O I S E ) → T H E
S Y S T E M H O V E R S A R O U N D I T
16
Asynchronous
Irregular (AI)
Brunel (2000)
AI

17. 17
Contact:
okinouchi@gmail.com
Ludmila Brochini
Ariadne A. Costa Maurício Girardi-Schappo Tawan T. A. Carvalho Mauro Copelli