# Chapters 6-7 (PPTX, 1904 KB) - URPP Foundations of Human ...

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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

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

Nagative

feedback

Positive
feedback

Mutual
inhibition

Input

Input

Input

Input

Two
-
neuron networks

General form (in absence of stimulus input):

𝑥
1

=


1
,

2

𝑥
2

=

(

1
,

2
)


1
.

2

as input to the update function


1
,

2
,


1
,

2



1
,

2
=
0



1
,

2
=
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

Equations
:

dB
dt
=
1
τ
B
(

B
+
L
1
+
A
)

dA
dt
=
1
τ
A
(

A
+
2B
)

B

A

L
ight

Equations
:



=
1
𝜏

(


+

1
+

)

=
1
𝜏

(

+
2
)

Nullclines
:


+

1
+

=
0

+
2
=
0

Equilibrium point:


𝑞
=

1
+
1
+
8
4

𝑞
=
2

Introduction to
Jacobian
:

Given

dt

1


𝑛
=

1
(

1
,

,

𝑛
)


𝑛
(

1
,

,

𝑛
)

Jacobian

𝜕


𝑥
1
,

,
𝑥
𝑛
𝜕
𝑥
1
,

,
𝑥
𝑛
=
𝜕

1
𝜕
𝑥
1

𝜕

1
𝜕
𝑥
𝑛

𝜕

𝑛
𝜕
𝑥
1

𝜕

𝑛
𝜕
𝑥
𝑛

Example: given
our update function


1
(

,

)
=
1
𝜏

(


+

1
+

)


(

,

)
=
1
𝜏

(

+
2
)

Jacobian

𝜕

1
𝜕
𝜕

1
𝜕
𝜕

𝜕
𝜕

𝜕
=

1
𝜏


1
𝜏


1
+

2
2
𝜏

1
𝜏

Linear stability

Proof:

Our equations



=


,

=

(

,

)

Apply a small perturbation to the steady state,
u,v

<< 1, take
this point as initial condition



0


𝑞
+

0

0

𝑞
+

0

Where

0
=

,

0
=

, u(t),v(t) represents deviation from

Proof (cont.):

Plug in and solve



(

)

=

(

𝑒𝑞
+


)

=

(

)

=



,

=

(

𝑞
+

,

𝑞
+

)



𝑞
,

𝑞
+

𝜕
𝜕


𝑞
,

𝑞
+

𝜕
𝜕

𝑞
,

𝑞
+

𝜕
𝜕

𝑞
,

𝑞
𝜕
𝜕

𝑞
,

𝑞



=

𝑒𝑞
+


=


=



,

=

(

𝑞
+

,

𝑞
+

)



𝑞
,

𝑞
+

𝜕

𝜕


𝑞
,

𝑞
+

𝜕

𝜕

𝑞
,

𝑞
+

𝜕

𝜕

𝑞
,

𝑞
𝜕

𝜕

𝑞
,

𝑞



Finally



=
𝜕

𝜕


𝑞
,

𝑞
𝜕

𝜕

𝑞
,

𝑞
𝜕

𝜕


𝑞
,

𝑞
𝜕

𝜕

𝑞
,

𝑞



Then use eigenvalue to determine asymptotic behavior

Negative feedback: a divisive gain control

Equations
:



=
1
10
(


+
10
1
+

)

=
1
10
(

+
2
)

Fixed point
(

𝑞
,

𝑞
)
=
(
2
,
4
)

Stability

analysis

Jacobian

at

(2,4)

=

1
1
0

1
25
1
5

1
10

Eigenvalues
𝜆
=

0
.
1
±
0
.
089
𝑖

=> asymptotically stable

Unique
stable

fixed

point

=>
our

fixed

point

is

a «global
attractor
»

Two
-
neuron
networks

Nagative

feedback

Positive
feedback

Mutual
inhibition

Input

Input

Input

Input

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

Equations:


1

=
1
𝜏


1
+

3

2


2

=
1
𝜏


2
+

3

1

E1
E2

A sigmoidal activation function:

=

100
𝑃
2
1
20
2
+
𝑃
2
0

0

<
0

P: stimulus strength

S: firing rate

A short
-
term memory circuit by positive
feedback

Equations:


1

=
1
𝜏


1
+

3

2


2

=
1
𝜏


2
+

3

1

Nullclines:


1
=

3

2
=
100
3

2
2
120
2
+
3

2
2


2
=

3

1
=
100
3

1
2
120
2
+
3

1
2

Equilibrium point:

9

1
3

900

1
2
+
120
2

1
=
0


1𝑞
=
0
,
20
,
80

E
2eq

can be obtained similarly

Equilibrium point:

(

1𝑞
,

2𝑞
)
=
0
,
0
,
20
,
20
,
80
,
80

Stability analysis:

(0,0
): Jacobian
=

0
.
05
0
0

0
.
05
,
𝜆
=

0
.
05
,

0
.
05

 

(20,20):
Jacobian

=

0
.
05
0
.
08
0
.
08

0
.
05
,
𝜆
=
+
0
.
03
,

0
.
13

𝑛  

(100,100):
Jacobian

=

0
.
05
0
.
02
0
.
02

0
.
05
,
𝜆
=

0
.
07
,

0
.
03

 

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

The steady states of E1 becomes


1
=
100
3

1
+

2
120
2
+
3

1
+

2

Demo

E1
E2
K

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

Nagative

feedback

Positive
feedback

Mutual
inhibition

Input

Input

Input

Input

Mutual inhibition: a winner
-
take
-
all neural
network for decision making


1

=
1
𝜏


1
+

𝐾
1

3

2


2

=
1
𝜏
(


2
+

𝐾
2

3

1
)

Demo

K1

E1
E2
K2

Chapter 6. Two
-
neuron networks

Nagative

feedback

Positive
feedback

Mutual
inhibition

Input

Input

Input

Input

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
-
input

T for target, D for distractor

𝜏
𝑇

=

T
+
S
(
E
T

3ND
)

𝜏

=

D
+

(


3
𝑁

1

3
)

E
T

T
D
E
D

D
E
D

Stimulus to target neuron:80, to disturbing neurons:79.8

Stimulus to target neuron: 80, to disturbing neurons: 79

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

𝜏
𝜕
𝑥
,

𝜕
=


(

)
+


(




(

)


𝐼
𝐼
(

)
+

(

)
)
𝑥
𝑥

𝜏
𝜕
𝐼
𝑥
,

𝜕
=

𝐼
(

)
+

𝐼
(


𝐼

(

)


𝐼
𝐼
𝐼
(

)
+

(

)
)
𝑥
𝑥

E(x,t
),
I(x,t
) := mean firing rates of neurons

x

:= position

P,Q := external inputs

w
EE
,

w
IE
,

w
EI
,

w
II
,

:= weights of interactions

𝜏
𝜕
𝑥
,

𝜕
=


+
1




(






𝐼
𝐼
+

)
𝑥
𝑥

𝜏
𝜕𝐼
𝑥
,

𝜕
=

𝐼
+
1

𝐼

𝐼
(


𝐼



𝐼𝐼
𝐼
+

)
𝑥
𝑥

S
patial
exponential
decay is determined by, e.g.







=


exp

(

𝑥

𝑥

𝜎
𝐸𝐸
)

x := position of input

x’ := position away from the input

Sigmoidal activation function

=
100
𝑃
2
𝜃
2
+
𝑃
2

P := stimulus input

Sigmoidal curve with respect to P

Example: short term memory in prefrontal cortex

A brief stimulus = 10ms, 100 µ
m

A brief stimulus = 10ms, 1000 µm

Wilson
-
Cowan model

Examples: short term memory, constant stimulus

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