Spikes, Decisions, Actions
The dynamical foundations of neuroscience
Valance WANG
Computational Biology and Bioinformatics, ETH Zurich
The last meeting
• Higher-dimensional linear dynamical systems
•
•
•
•
General solution
Asymptotic stability
Oscillation
Delayed feedback
• Approximation and simulation
Outline
• Chapter 6. Nonlinear dynamics and bifurcations
• Two-neuron networks
• Negative feedback: a divisive gain control
• Positive feedback: a short term memory circuit
• Mutual Inhibition: a winner-take-all network
• Stability of steady states
• Hysteresis and Bifurcation
• Chapter 7. Computation by excitatory and inhibitory networks
• Visual search by winner-take-all network
• Short term memory by Wilson-Cowan cortical dynamics
Chapter 6. Two-neuron networks
Input
Input
Nagative feedback
Input
Positive feedback
Input
Mutual inhibition
Two-neuron networks
• General form (in absence of stimulus input):
•
•
𝑑𝑥1
𝑑𝑡
𝑑𝑥2
𝑑𝑡
= 𝐹 𝑥1 , 𝑥2
= 𝐺(𝑥1 , 𝑥2 )
• Reading current state 𝑥1 . 𝑥2 as input to the update function
𝐹 𝑥1 , 𝑥2 , 𝐺 𝑥1 , 𝑥2
• Steady states:
• 𝐹 𝑥1 , 𝑥2 = 0
• 𝐺 𝑥1 , 𝑥2 = 0
Negative feedback: a divisive gain control
• In retina,
• Light -> Photo-receptors -> Bipolar cells -> Ganglion cells -> optic
nerves
•
Amacrine cell
• This forms a relay chain of information
• To stabilize representation of information, bipolar cells receive
negative feedback from amacrine cell
Negative feedback: a divisive gain control
• In retina,
Negative feedback: a divisive gain control
Light
B
A
• Equations:
•
dB
dt
=
1
(−B
τB
+
L
)
1+A
•
dA
dt
=
1
(−A
τA
+ 2B)
• Equations:
𝑑𝐵
𝑑𝑡
•
𝑑𝐴
𝑑𝑡
=
1
(−𝐵
𝜏𝐵
+
𝐿
)
1+𝐴
=
1
(−𝐴
𝜏𝐴
+ 2𝐵)
8
• Nullclines:
• −𝐵 +
𝐿
1+𝐴
=0
• −𝐴 + 2𝐵 = 0
• Equilibrium point:
• 𝐵𝑒𝑞 =
−1+ 1+8𝐿
4
• 𝐴𝑒𝑞 = 2𝐵
dB/dt=0
dA/dt=0
9
A - amacrine cell response
•
phase plane analysis for L=10
10
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
B - bipolar cell response
7
8
9
10
Linear stability of steady states
• Introduction to Jacobian:
𝑥1
𝐹1 (𝑥1 , … , 𝑥𝑛 )
𝑑
… =
…
• Given
dt
𝑥𝑛
𝐹𝑛 (𝑥1 , … , 𝑥𝑛 )
• Jacobian ≡
𝜕𝐹 𝑥1 ,…,𝑥𝑛
𝜕 𝑥1 ,…,𝑥𝑛
𝜕𝐹1
𝜕𝑥1
…
= …
𝜕𝐹1
𝜕𝑥𝑛
…
𝜕𝐹𝑛
𝜕𝑥1
…
𝜕𝐹𝑛
𝜕𝑥𝑛
• Example: given our update function
1
𝐿
• 𝐹1 (𝐵, 𝐴) = 𝜏 (−𝐵 + 1+𝐴)
𝐵
1
• 𝐺(𝐵, 𝐴) = 𝜏 (−𝐴 + 2𝐵)
𝐴
• Jacobian ≡
𝜕𝐹1
𝜕𝐵
𝜕𝐺
𝜕𝐵
𝜕𝐹1
𝜕𝐴
𝜕𝐺
𝜕𝐴
1
=
−𝜏
2
𝜏𝐴
𝐵
1
𝐿
2
𝐵 1+𝐴
1
−𝜏
𝐴
−𝜏
Linear stability of steady states
Linear stability of steady states
• Proof:
• Our equations
•
•
𝑑𝐵
𝑑𝑡
𝑑𝐴
𝑑𝑡
= 𝐹 𝐵, 𝐴
= 𝐺(𝐵, 𝐴)
• Apply a small perturbation to the steady state, u,v << 1, take
this point as initial condition
• 𝐵 0 ≔ 𝐵𝑒𝑞 +𝑢 0
• 𝐴 0 ≔ 𝐴𝑒𝑞 +𝑣 0
• Where 𝑢 0 = 𝑢, 𝑣 0 = 𝑣, u(t),v(t) represents deviation from
steady states
• Proof (cont.):
• Plug in and solve
𝑑(𝐵𝑒𝑞 +𝑢 𝑡 )
•
𝑑 𝐵(𝑡)
𝑑𝑡
=
•
𝑑𝐴
𝑑𝑡
𝑑 𝐴𝑒𝑞 +𝑣 𝑡
𝑑𝑢(𝑡)
=
= 𝐹 𝐵, 𝐴 = 𝐹(𝐵𝑒𝑞 + 𝑢, 𝐴𝑒𝑞 + 𝑣)
𝑑𝑡
𝑑𝑡
𝜕𝐹
𝜕𝐹
≈ 𝐹 𝐵𝑒𝑞 , 𝐴𝑒𝑞 + 𝑢
𝐵 ,𝐴
+𝑣
𝐵 ,𝐴
+⋯
𝜕𝐵 𝑒𝑞 𝑒𝑞
𝜕𝐴 𝑒𝑞 𝑒𝑞
𝜕𝐹
𝜕𝐹
𝑢
≈
𝐵 ,𝐴
𝐵 ,𝐴
𝜕𝐵 𝑒𝑞 𝑒𝑞
𝜕𝐴 𝑒𝑞 𝑒𝑞 𝑣
=
𝑑𝑡
=
𝑑𝑣 𝑡
𝑑𝑡
𝜕𝐺
≈ 𝐺 𝐵𝑒𝑞 , 𝐴𝑒𝑞 + 𝑢
𝜕𝐵
𝜕𝐺
𝜕𝐺
≈
𝐵𝑒𝑞 , 𝐴𝑒𝑞
𝜕𝐵
𝜕𝐴
= 𝑑𝐺 𝐵, 𝐴 = 𝐺(𝐵𝑒𝑞 + 𝑢, 𝐴𝑒𝑞 + 𝑣)
𝜕𝐺
𝐵𝑒𝑞 , 𝐴𝑒𝑞 + 𝑣
𝐵 ,𝐴
+⋯
𝜕𝐴 𝑒𝑞 𝑒𝑞
𝑢
𝐵𝑒𝑞 , 𝐴𝑒𝑞 𝑣
• Finally
•
𝑑
𝑑𝑡
𝐵 =
𝐴
𝜕𝐹
𝜕𝐵
𝜕𝐺
𝜕𝐵
𝐵𝑒𝑞 , 𝐴𝑒𝑞
𝐵𝑒𝑞 , 𝐴𝑒𝑞
𝜕𝐹
𝜕𝐴
𝜕𝐺
𝜕𝐴
𝐵𝑒𝑞 , 𝐴𝑒𝑞
𝐵𝑒𝑞 , 𝐴𝑒𝑞
𝑢
𝑣
• Then use eigenvalue to determine asymptotic behavior
Negative feedback: a divisive gain control
• Equations:
•
𝑑𝐵
𝑑𝑡
𝑑𝐴
𝑑𝑡
=
=
1
10
(−𝐵 +
)
10
1+𝐴
1
(−𝐴 + 2𝐵)
10
• Fixed point (𝐵𝑒𝑞, 𝐴𝑒𝑞 ) = (2,4)
• Stability analysis
• Jacobian at (2,4)
=
1
10
1
5
−
1
25
1
−
10
−
dB/dt=0
dA/dt=0
9
8
A - amacrine cell response
•
phase plane analysis for L=10
10
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
B - bipolar cell response
7
8
9
10
• Eigenvalues 𝜆 = −0.1 ± 0.089𝑖 => asymptotically stable
• Unique stable fixed point => our fixed point is a «global attractor»
Two-neuron networks
Input
Input
Nagative feedback
Input
Positive feedback
Input
Mutual inhibition
A short-term memory circuit by positive
feedback
• In monkeys’ prefrontal cortex
A short-term memory circuit by positive
feedback
• First, let’s analyze the behavior of the system in absence of
external stimulus
E1
E2
• Equations:
•
•
𝑑𝐸1
𝑑𝑡
𝑑𝐸2
𝑑𝑡
=
=
1
𝜏
1
𝜏
−𝐸1 + 𝑆 3𝐸2
−𝐸2 + 𝑆 3𝐸1
• A sigmoidal activation function: 𝑆 𝑃 =
• P: stimulus strength
• S: firing rate
100𝑃2
1202 +𝑃2
0
𝑃≥0
𝑃<0
A short-term memory circuit by positive
feedback
• Equations:
•
•
𝑑𝐸1
𝑑𝑡
𝑑𝐸2
𝑑𝑡
=
=
1
𝜏
1
𝜏
−𝐸1 + 𝑆 3𝐸2
−𝐸2 + 𝑆 3𝐸1
• Nullclines:
• 𝐸1 = 𝑆 3𝐸2 =
100 3𝐸2 2
1202 + 3𝐸2 2
• 𝐸2 = 𝑆 3𝐸1 =
100 3𝐸1 2
1202 + 3𝐸1 2
• Equilibrium point:
• 9𝐸13 − 900𝐸12 + 1202 𝐸1 = 0
• ⇒ 𝐸1𝑒𝑞 = 0,20,80
• E2eq can be obtained similarly
phase plane analysis
100
dE1/dt=0
dE2/dt=0
90
80
70
E2
60
50
40
30
20
10
0
0
10
20
30
40
50
E1
60
70
80
90
100
• Equilibrium point:
• (𝐸1𝑒𝑞 , 𝐸2𝑒𝑞 ) = 0,0 , 20,20 , 80,80
• Stability analysis:
−0.05
0
, 𝜆 = −0.05, −0.05 ⇒ 𝑠𝑡𝑎𝑏𝑙𝑒
0
−0.05
−0.05 0.08
• (20,20): Jacobian =
, 𝜆 = +0.03, −0.13 ⇒
0.08 −0.05
𝑢𝑛𝑠𝑡𝑎𝑏𝑙𝑒
• (0,0): Jacobian =
• (100,100): Jacobian =
𝑠𝑡𝑎𝑏𝑙𝑒
−0.05
0.02
0.02
, 𝜆 = −0.07, −0.03 ⇒
−0.05
Hysteresis and Bifurcation
• The term ‘hysteresis’ is derived from Greek, meaning ‘to lag
behind’.
• In present context, this means that the present state of our
neural network is determined not just by the present state
and input, but also by the state and input in the history
(“path-dependent”).
Hysteresis and Bifurcation
• Suppose we apply a brief stimulus K to the neural network
K
E1
E2
• The steady states of E1 becomes
• 𝐸1 =
• Demo
100 3𝐸1 +𝐾 2
1202 + 3𝐸1 +𝐾 2
Hysteresis and Bifurcation
• Due to change in parameter value K, a pair of equilibrium
points may appear or disappear. This phenomenon is known
as bifurcation.
Two-neuron networks
Input
Input
Nagative feedback
Input
Positive feedback
Input
Mutual inhibition
Mutual inhibition: a winner-take-all neural
network for decision making
•
•
K1
K2
E1
E2
𝑑𝐸1
𝑑𝑡
𝑑𝐸2
𝑑𝑡
=
=
• Demo
1
−𝐸1
𝜏
1
(−𝐸2
𝜏
+ 𝑆 𝐾1 − 3𝐸2
+ 𝑆 𝐾2 − 3𝐸1 )
Chapter 6. Two-neuron networks
Input
Input
Nagative feedback
Input
Positive feedback
Input
Mutual inhibition
Chapter 7. Multiple-Neuron-network
• Visual search by a winner-take-all network
• Wilson-Cowan cortical dynamics
Visual search by winner-take-all network
• Visual search
Visual search by winner-take-all network
• A N+1 Neuron-network, each neuron receives perceptive
input
• T for target, D for distractor
ET
ED
ED
T
D
D
𝑑𝑇
𝑑𝑡
𝑑𝐷
𝜏
𝑑𝑡
• 𝜏
= −T + S(ET − 3ND)
•
= −D + 𝑆(𝐸𝐷 − 3 𝑁 − 1 𝐷 − 3𝑇)
• Stimulus to target neuron:80, to disturbing neurons:79.8
35
winner neuron
30
response
25
20
15
10
5
0
0
100
200
300
400
500
time
600
700
800
900
1000
• Stimulus to target neuron: 80, to disturbing neurons: 79
35
winner neuron
30
response
25
20
15
10
5
0
0
100
200
300
400
500
time
600
700
800
900
1000
• Further, this model can be extrapolated for higher level
cognitive decisions. It is common experience that decisions
are more difficult to make and take longer when the number
of appealing alternatives increases.
• Once a decision is definitely made, however, humans are
reluctant to change their decision. (Hysteresis in cognitive
process!)
Wilson-Cowan model (1973)
• Cortical neurons may be divided into two classes:
• excitatory (E), usu. Pyramidal neurons
• and inhibitory (I), usu. interneurons
• All forms of interaction occur between these classes:
• E -> E, E -> I, I -> E, I -> I
• Recurrent excitatory network are local, while inhibitory
connections are long range
• A one-dimensional spatial-temporal model
𝜕𝐸 𝑥,𝑡
𝜕𝑡
𝜕𝐼 𝑥,𝑡
𝜏
𝜕𝑡
• 𝜏
= −𝐸(𝑥) + 𝑆𝐸 (
•
= −𝐼(𝑥) + 𝑆𝐼 (
•
•
•
•
𝑥 𝑤𝐸𝐸 𝐸(𝑥)
𝑥 𝑤𝐸𝐼 𝐸(𝑥)
−
−
𝑥 𝑤𝐼𝐸 𝐼(𝑥)
𝑥 𝑤𝐼𝐼 𝐼(𝑥)
E(x,t), I(x,t) := mean firing rates of neurons
x := position
P,Q := external inputs
wEE, wIE, wEI, wII, := weights of interactions
+ 𝑃(𝑥))
+ 𝑄(𝑥))
𝜕𝐸 𝑥,𝑡
𝜕𝑡
𝜕𝐼 𝑥,𝑡
𝜏
𝜕𝑡
• 𝜏
= −𝐸 + 1 − 𝑘𝐸 𝑆𝐸 (
•
= −𝐼 + 1 − 𝑘𝐼 𝑆𝐼 (
𝑥 𝑤𝐸𝐸 𝐸
𝑥 𝑤𝐸𝐼 𝐸
−
−
𝑥 𝑤𝐼𝐸 𝐼
𝑥 𝑤𝐼𝐼 𝐼
+ 𝑄)
• Spatial exponential decay is determined by, e.g.
• 𝑤𝐸𝐸 𝑥 −
𝑥′
= 𝑏𝐸𝐸 exp(−
𝑥−𝑥 ′
𝜎𝐸𝐸
)
• x := position of input
• x’ := position away from the input
• Sigmoidal activation function
• 𝑆 𝑃 =
100𝑃2
𝜃 2 +𝑃2
• P := stimulus input
• Sigmoidal curve with respect to P
+ 𝑃)
• Example: short term memory in prefrontal cortex
• A brief stimulus = 10ms, 100 µm
110
100
E (red) & I (blue) Responses
90
80
70
60
50
40
30
20
10
0
0
200
400
600
800 1000 1200
Distance in microns
1400
800 1000 1200
Distance in microns
1400
1600
1800
2000
1600
1800
2000
• A brief stimulus = 10ms, 1000 µm
110
100
E (red) & I (blue) Responses
90
80
70
60
50
40
30
20
10
0
0
200
400
600
Wilson-Cowan model
• Examples: short term memory, constant stimulus
110
100
E (red) & I (blue) Responses
90
80
70
60
50
40
30
20
10
0
0
500
1000
1500
2000
2500
Distance in microns
3000
3500
4000
Summary of Chapter 7
• Winner-take all network
• Visual search can be disturbed by the number of irrelevant but
similar objects
• Wilson-Cowan model
• A one-dimensional spatial-temporal dynamical system
• Applications:
• Short term memory in prefrontal cortex