A mathematical approach to retinal waves

HAL Id: cel-01626745

https://hal.inria.fr/cel-01626745

Submitted on 31 Oct 2017

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Bruno Cessac, Dora Karvouniari

To cite this version:

Bruno Cessac, Dora Karvouniari. A mathematical approach to retinal waves. Doctoral. Chile. 2017.

�cel-01626745�

A mathematical approach to

retinal waves

B. Cessac, D. Karvouniari

Biovision team, INRIA, Sophia

Antipolis

Acknowledgement

L. Gil, INLN, Sophia Antipolis

O. Marre, Institut de la Vision, Paris

S. Picaud, Institut de la Vision, Paris

Matthias Hennig, University of Edinburgh

Evelyne Sernagor, Newcastle University

2

Visual

cortex

LGN

Visual system and retinal waves

Retinal structure

3

Visual

cortex

Retinal structure

LGN

Visual system and retinal waves

Spontaneous spatio-temporal ‘’waves’’

during development

§

  Disappear short after birth

Spatio-Temporal evolution of a

retinal wave in mice during

development, Maccione et al. 2014

4

5

Stages of Retinal Waves During

Development

Stage I

•

Formation of

retina circuitry

• 

Chemical

synapses not

formed yet

• 

Gap juction-mediated

6

Stages of Retinal Waves During

Development

Stage I

Stage II

•

Formation of

retina circuitry

• 

Chemical

synapses not

formed yet

• 

Gap juction-mediated

•

Retinotopic mapping

• 

Nicotinic Acetylcholine Receptors

(nAChR)

7

Stages of Retinal Waves During

Development

Stage I

Stage II

Stage III

•

Formation of

retina circuitry

• 

Chemical

synapses not

formed yet

• 

Gap juction-mediated

•

Retinotopic mapping

• 

Nicotinic Acetylcholine Receptors

(nAChR)

•

Disappear when

vision is functional

• 

Glutamate –

AMPA receptors

groups.oist.jp/dnu/

8

Focus on stage II retinal waves

Stage II

(cholinergic

retinal waves)

Zheng et al. 2006

Stage II retinal waves are well studied

experimentally and there is already existing

work on their modelling

Starburst Amacrine

Cell (SAC)

Goal of this lecture

Idea:

Identify generic mechanisms generating stage II

retinal waves and analyze them mathematically.

Goal of this lecture

10

Motivation

:

•  Retinal waves instruct the shaping of the visual

system

Idea:

Identify generic mechanisms generating stage II

retinal waves and analyze them mathematically.

Goal of this lecture

11

Motivation

:

•  Retinal waves instruct the shaping of the visual

system

•  Understanding the mechanisms that generate

them may help to control them

•  New mathematical problems and new techniques.

Idea:

Identify generic mechanisms generating stage II

retinal waves and analyze them mathematically.

12

12

Experiments for the emergence of retinal waves

Experiment for isolated neurons,

Zheng et al., 2006, Nature

Retinal waves require three components:

i)

_{Spontaneous bursting activity}

13

13

Experiment for isolated neurons,

Zheng et al., 2006, Nature

Retinal waves require three components:

i)

_{Spontaneous bursting activity}

ii)

_{Refractory mechanism (}

slow After HyperPolarisation- sAHP

)

14

14

Experiment for coupled and isolated

neurons, Zheng et al., 2006, Nature

Retinal waves require three components:

i)

_{Spontaneous bursting activity}

ii)

_{Refractory mechanism (}

slow After HyperPolarisation- sAHP

)

iii)

_{Coupling (through Acetylcholine neurotransmitter)}

Coupled neurons

synchronize

Coupled neurons

synchronize

Experiments for the emergence of retinal waves

15

15

Experiment for coupled and isolated

neurons, Zheng et al., 2006, Nature

Retinal waves require three components:

i)

_{Spontaneous bursting activity}

ii)

_{Refractory mechanism (}

slow After HyperPolarisation- sAHP

)

iii)

_{Coupling (through Acetylcholine neurotransmitter)}

Coupled neurons

synchronize

Isolated Neurons burst

independently

Coupled neurons

synchronize

Experiments for the emergence of retinal waves

From the non linear physics perspective a retinal wave is

a spatiotemporal activity which:

•  is generated by the conjunction of

local (cell level)

nonlinear characteristics and network effects

16

Mathematical characterization

of Retinal Waves

From the non linear physics perspective a retinal wave is

a spatiotemporal activity which:

•  is generated by the conjunction of

local (cell level)

nonlinear characteristics and network effects

•  can have different

forms

(fronts, spiral, standing

waves, spatio-temporal chaos)

17

Mathematical characterization

of Retinal Waves

From the non linear physics perspective a retinal wave is

a spatiotemporal activity which:

•  is generated by the conjunction of

local (cell level)

nonlinear characteristics and network effects

•  can have different

forms

(fronts, spiral, standing

waves, spatio-temporal chaos)

•  is

induced by generic mechanisms

that can be

studied in the context of dynamical systems theory

18

Mathematical characterization

of Retinal Waves

From the non linear physics perspective a retinal wave is

a spatiotemporal activity which:

•  is generated by the conjunction of

local (cell level)

nonlinear characteristics and network effects

•  can have different

forms

(fronts, spiral, standing

waves, spatio-temporal chaos)

•  is

induced by generic mechanisms

that can be

studied in the context of dynamical systems theory

•  can be associated with

bifurcations

19

Mathematical characterization

of Retinal Waves

•  Model

biophysically

the dynamics of

isolated neurons

(SACs)

•  Carefully model all currents involved

•  Fix parameters from experiments

20

Our strategy

•  Model

biophysically

the dynamics of

isolated neurons

(SACs)

•  Carefully model all currents involved

•  Fix parameters from experiments

•

Bifurcation analysis

of the non-linear dynamical

system for isolated neurons

21

Our strategy

•  Model

biophysically

the dynamics of

isolated neurons

(SACs)

•  Carefully model all currents involved

•  Fix parameters from experiments

•

Bifurcation analysis

of the non-linear dynamical

system for isolated neurons

•  Extract a

generic biophysical mechanism

of

spontaneous bursting activity for SACs

22

Our strategy

•  Model

biophysically

the dynamics of

isolated neurons

(SACs)

•  Carefully model all currents involved

•  Fix parameters from experiments

•

Bifurcation analysis

of the non-linear dynamical

system for isolated neurons

•  Extract a

generic biophysical mechanism

of

spontaneous bursting activity for SACs

•  Model the network interactions:

•  Study the effects of synaptic coupling to

synchrony and wave initiation

•  Search for biophysical parameters which control

spatiotemporal patterns

23

Our strategy

Modelling the bursting of individual Starburst

Amacrine Cells

Bruno Cessac and Dora Karvouniari

Biovision Team, INRIA Sophia Antipolis,France.

1

From biophysics to modelling

2

Model analysis

Biophysics of bursting SAC

Retinal waves require three components:

Spontaneous bursting activity

Refractory mechanism (slow After HyperPolarisation- sAHP)

Coupling (through Acetylcholine neurotransmitter)

Biophysics of bursting SAC

Figure:

Zheng et al., Nature, 2006. Grey curve: Fast oscillations due to

voltage-gated Ca

+2

_{channels and Ca}

+2

_{dependent K}

+

_{channels. Black}

Biophysics of bursting SAC























C

m

dV

_dt

= −g

L

(V − V

L

) − g

C

M

∞

(V )(V − V

C

) − g

K

N(V − V

K

)

+I

ex

(•)

τ

N

dN

_dt

= Λ(V )(N

∞

(V ) − N)

(1)

Morris-Lecar model with fast Potassium.

Dynamics









































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























C

m

dV

_dt

= −g

L

(V − V

L

) − g

C

M

∞

(V )(V − V

C

) − g

K

N(V − V

K

)

−g

sAHP

R

4

(V − V

K

)

τ

N

dN

_dt

= Λ(V )(N

∞

(V ) − N)

τ

C

dC

_dt

= −

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

τ

S

dS

_dt

= α

S

(1 − S)C

4

− S

τ

R

dR

_dt

= α

R

S

(1 − R) − R

(2)

Dynamics







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

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









































C

m

dV

_dt

= −g

L

(V − V

L

) − g

C

M

∞

(V )(V − V

C

) − g

K

N(V − V

K

)

−g

sAHP

R

4

G

sAHP

(R)(V − V

K

)

τ

N

dN

_dt

= Λ(V )(N

∞

(V ) − N)

τ

C

dC

_dt

= −

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

τ

S

dS

_dt

= α

S

(1 − S)C

4

− S

τ

R

dR

_dt

= α

R

S

(1 − R) − R

(3)

Dynamics

Leak time scale: τ

L

=

_g

C

.

Conductances rescaling: ˜

g

X

=

g

_g

.





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

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

















τ

L

dV

_dt

= −˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

τ

N

dN

_dt

= Λ(V )(N

∞

(V ) − N)

τ

C

dC

_dt

= −

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

τ

S

dS

_dt

= α

S

(1 − S)C

4

− S

τ

R

dR

_dt

= α

R

S

(1 − R) − R

Multi-scale dynamics

Fast V , N. τ

L

= 11 ms, τ

N

= 5 ms.

Medium C . τ

C

= 2 s.

Fast-scale

t

f

=

τ

t

⇒ dt

= τ

L

dt

f







































































dV

dt

= −˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

τ

τ

dN

dt

= Λ(V )(N

∞

(V ) − N)

τ

τ

dC

dt

= −

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

τ

τ

dS

dt

= α

S

(1 − S)C

4

_{− S}

τ

τ

dR

dt

= α

R

S

(1 − R) − R

Fast-scale

Set ǫ

X

=

τ

τ

, X = C , R, S.

ǫ

C

=

11

₂

×

10

−

3

∼

5.5 × 10

−

3

,

ǫ

S

= ǫ

R

=

11

₄₄

×

10

−

3

∼

2.5 × 10

−

4























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

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













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























dV

dt

= −˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

dN

dt

=

τ

τ

Λ(V )(N

∞

(V ) − N)

dC

dt

= ǫ

C

h

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

i

dS

dt

= ǫ

S

 α

S

(1 − S)C

4

_{− S}



dR

dt

= ǫ

R

[ α

R

S(1 − R) − R ]

Fast-scale: approximation



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



























































dV

dt

= −˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

dN

dt

=

τ

τ

Λ(V )(N

∞

(V ) − N)

dC

dt

= 0

dS

dt

= 0

dR

dt

= 0

Morris-Lecar model analysis



















dV

dt

= −˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

dN

dt

=

τ

τ

Λ(V )(N

∞

(V ) − N)

(4)

Morris-Lecar model analysis







dV

dt

= −˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

) + I

ext

dN

dt

=

τ

τ

Λ(V )(N

∞

(V ) − N)

(5)

Bifurcation analysis when varying the parameter I

ext

.

Morris-Lecar model analysis

The saddle-node bifurcation

Figure:

From https://elmer.unibas.ch/pendulum/bif.htm and

The Hopf bifurcation

Figure:

From Navarro et al, ”Control of the Hopf bifurcation in the

Takens-Bogdanov bifurcation”, CDC2008

The homoclinic bifurcation

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Karvouniari et al, ICMNS, 2016

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Back to Zhou experiment

Figure:

Zheng et al., Nature, 2006. Grey curve: Fast oscillations due to

voltage-gated Ca

+2

_{channels and Ca}

+2

_{dependent K}

+

_{channels. Black}

Back to Zhou experiment

Back to Zhou experiment

Karvouniari et al, submitted to Plos Comp. Bio.

0 50 100 150 Time (ms) −50 0 V (mV) Oscillations Block Ca++ Block K+

Figure:

when Ca+2related conductances are set to zero. Red. Blocking the oscillations upon setting the voltage-gated K + conductance to zero. Right: Zheng et al. 2006 experimental figure. Grey curve: Fast oscillations due to voltage-gated Ca+2channels and Ca+2dependent K+channels. Black curve: Application of Cd+blocking Ca+2 related channels.

Dynamics of bursting



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



















τ

L

dV

_dt

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

τ

N

dN

_dt

=

Λ(V )(N

∞

(V ) − N)

τ

C

dC

dt

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

τ

S

dS

_dt

=

α

S

(1 − S)C

4

− S

τ

R

dR

_dt

=

α

R

S(1 − R) − R

Fast V , N. τ

L

= 11 ms, τ

N

= 5 ms.

Medium C . τ

C

= 2 s.

Slow S , R. τ

R

= τ

S

= 44 s.

Slow time-scale

t

s

=

τ

t

⇒ dt

= τ

S

dt

s



























































τ

τ

dV

dt

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

τ

τ

dN

dt

=

Λ(V )(N

∞

(V ) − N)

τ

τ

dC

dt

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

dS

dt

=

α

S

(1 − S)C

4

_{− S}

dR

dt

=

α

R

S(1 − R) − R

Slow time-scale

ǫ

V

=

τ

τ

∼

2.5 × 10

−

4

,

ǫ

N

=

τ

τ

∼

1 × 10

−

3

ǫ

C

=

τ

τ

∼

4.5 × 10

−

2



























































ǫ

V

dV

_dt

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

ǫ

N

dN

_dt

=

Λ(V )(N

∞

(V ) − N)

ǫ

C

dC

_dt

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

dS

dt

=

α

S

(1 − S)C

4

_{− S}

dR

dt

=

α

R

S(1 − R) − R

Slow time-scale approximation



























































0

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

_C

) − ˜

g

_K

N(V − V

_K

)

− ˜

G

sAHP

(R)(V − V

K

)

0

=

Λ(V )(N

∞

(V ) − N)

0

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

dS

dt

=

α

S

(1 − S)C

4

_{− S}

dR

dt

=

α

R

S

(1 − R) − R

Steady state

V

=

V

ML

(V ) + ˜

G

sAHP

(R)V

K

˜

g

ML

(V ) + ˜

G

sAHP

(R)

where:

V

ML

(V ) = V

L

+ ˜

g

C

M

∞

(V )V

C

+ ˜

g

K

N

∞

(V )V

K

;

˜

g

ML

(V )

= 1 + ˜

g

C

M

∞

(V ) + ˜

g

K

N

∞

(V ),

Steady state

V

follows ”adiabatically” R.

V ≡ f

(V , R) =

V

ML

(V ) + ˜

G

sAHP

(R)V

K

˜

g

ML

(V ) + ˜

G

sAHP

(R)

(6)

Steady state

V

follows ”adiabatically” R.

V ≡ f

(V , R) =

V

ML

(V ) + ˜

G

sAHP

(R)V

K

˜

g

ML

(V ) + ˜

G

sAHP

(R)

(6)

Implicit function theorem.

Wherever

∂f

∂V

6= 1 V follows a unique and differentiable branch

V ≡ V

(R). At points R where

∂f

∂V

= 1 this branch may disappear

Hint

V ≡ f

(V , R) =

V

ML

(V ) + ˜

G

sAHP

(R)V

K

˜

g

ML

(V ) + ˜

G

sAHP

(R)

(7)

Taking the differential gives:

dV



1 −

∂f

∂V



=

∂f

∂R

dR

⇒

dV

dR

=

∂f

∂R

1 −

_∂V

∂f

Morris-Lecar model analysis

Medium-scale

t

m

=

τ

t

⇒ dt

= τ

C

dt

m



























































τ

τ

dV

dt

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

τ

τ

dN

dt

=

Λ(V )(N

∞

(V ) − N)

dC

dt

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

τ

τ

dS

dt

=

α

S

(1 − S)C

4

_{− S}

τ

τ

dR

dt

=

α

R

S

(1 − R) − R

Medium-scale

ǫ

V

=

τ

τ

∼

5 × 10

−

3

,

ǫ

N

=

τ

τ

∼

2.5 × 10

−

3

ǫ

S

= ǫ

R

=

τ

τ

∼

4.5 × 10

−

2



























































ǫ

V

_dt

dV

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − ˜

g

K

N(V − V

K

)

− ˜

G

sAHP

(R)(V − V

K

)

ǫ

N

_dt

dN

=

Λ(V )(N

∞

(V ) − N)

dC

dt

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

dS

dt

=

ǫ

S

 α

S

(1 − S)C

4

_{− S}



dR

dt

=

ǫ

R

[ α

R

S

(1 − R) − R ]

Medium-scale approximation























































0

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

_C

) − ˜

g

_K

N(V − V

_K

)

− ˜

G

sAHP

(R)(V − V

K

)

0

=

Λ(V )(N

∞

(V ) − N)

dC

dt

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

dS

dt

=

0

dR

dt

=

0

Medium-scale approximation

At this time scale:

R, S

are constant;

V

follows the variations of R;

dC

dt

= −

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

Medium-scale approximation

Rapid (medium time scale between upper/lower branch) ”Outer

dynamics”.

Slow time-scale approximation



























































0

=

−˜

g

L

(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

_C

) − ˜

g

_K

N(V − V

_K

)

− ˜

G

sAHP

(R)(V − V

K

)

0

=

Λ(V )(N

∞

(V ) − N)

0

=

−

H

α

C

+ C

0

− δ

C

g

C

M

∞

(V )(V − V

C

)

dS

dt

=

α

S

(1 − S)C

4

_{− S}

dR

dt

=

α

R

S

(1 − R) − R

”Inner dynamics”

V

follows R;

C

follows V ;

Note P(R) ≡ C

4

_{, the average value of C}

4

_{for a given value of}

R.

⇒

Dynamics on the slow branches.















dR

dt

= α

R

S

(1 − R) − R

dS

dt

= α

S

(1 − S)P(R) − S

”Inner dynamics”

Nullclines.

R

Nullcline:

dR

dt

= 0 ⇒

The vector field is vertical on the line S =

α

_(1−R)

R

S

Nullcline:

_dt

dS

s

= 0 ⇒

The vector field is horizontal on the line S =

α

P(R)

Conclusions

Conclusions

We have now a model reproducing individual SACs bursting.

Here SACs burst periodically with a frequency determined by

the parameters of the model (mainly τ

R

).

Conclusions

We have now a model reproducing individual SACs bursting.

Here SACs burst periodically with a frequency determined by

the parameters of the model (mainly τ

R

).

A small change of parameters make them into a rest state

where they burst only upon an external excitation (noise).

Conclusions

We have now a model reproducing individual SACs bursting.

Here SACs burst periodically with a frequency determined by

the parameters of the model (mainly τ

R

).

A small change of parameters make them into a rest state

where they burst only upon an external excitation (noise).

What happens when they cells are coupled with

Acetylcholine ?

Modelling stage II retinal waves

Bruno Cessac and Dora Karvouniari

Biovision Team, INRIA Sophia Antipolis,France.

15-01-2017

1

Coupling SACs with Acetylcholine

2

Dynamical systems analysis

3

A simplified setting to mathematically study retinal waves

4

Generalisations

Coupling SACs with Acetylcholine

Coupling SACs with Acetylcholine

Coupling SACs with Acetylcholine

Figure:

Nicotinic receptors and conductance.

Coupling SACs with Acetylcholine

Coupling SACs with Acetylcholine

Figure:

Equation of membrane potential.

Main currents

G

sAHP

(R

i

) = g

sAHP

R

i

4

,

(1)

sAHP conductance,

G

A

({A

k

}

k∈B

) = g

A

X

k∈B

U(A

k

),

(2)

Ach synaptic conductance with:

U

(A) =

A

2

γ

A

+ A

2

,

(3)

B

i

is the set of index of neurons connected to i.

Dynamics























































































C

m

dV

_dt

= −g

L

(V

i

− V

L

) − g

C

M

∞

(V

i

)(V

i

− V

C

) − g

K

N

i

(V

i

− V

K

)

−G

sAHP

(R

i

)(V

i

− V

K

) − G

A

({A

k

}

k∈B

)(V

i

− V

A

)

τ

N

dN

_dt

= Λ(V

i

)(N

∞

(V

i

) − N

i

)

τ

C

dC

_dt

= −

_H

α

C

i

+ C

0

− δ

C

g

C

M

∞

(V

i

)(V

i

− V

C

)

τ

S

dS

_dt

= α

S

(1 − S

i

)C

i

4

− S

i

τ

R

dR

_dt

= α

R

S

i

(1 − R

i

) − R

i

d A

dt

= −µA

i

+ β

A

T

A

(V

i

),

(4)

Dynamics

Leak time scale: τ

L

=

_g

C

. Ach time scale: τ

A

=

_µ

1

.

Conductances rescaling: ˜

g

X

=

g

_g

.











































































τ

L

dV

_dt

=

−(V

i

− V

L

) − ˜

g

C

M

∞

(V

i

)(V

i

− V

C

) − ˜

g

K

N

i

(V

i

− V

K

)

− ˜

G

sAHP

(R

i

)(V

i

− V

K

) − ˜

G

A

({A

k

}

k∈B

)(V

i

− V

A

);

τ

N

dN

_dt

=

Λ(V

i

)(N

∞

(V

i

) − N

i

);

τ

C

dC

_dt

=

−

_H

α

C

i

+ C

0

− δ

C

g

C

M

∞

(V

i

)(V

i

− V

C

);

τ

S

dS

_dt

=

α

S

(1 − S

i

)C

i

4

− S

i

;

τ

R

dR

_dt

=

α

R

S

i

(1 − R

i

) − R

i

;

τ

A

d A

_dt

=

−A

i

+

β

_µ

T

A

(V

i

);

Multi-scale dynamics

Fast V , N. τ

L

= 11 ms, τ

N

= 5 ms.

Medium C , A. τ

C

= 2 s, τ

A

= 1.86 s.

A simplified setting to mathematically study retinal waves

Bruno Cessac and Dora Karvouniari Modelling stage II retinal waves

A simplified setting to mathematically study retinal waves

Main hypotheses:

1) We neglect the fast Potassium current. V

i

has only two stable states:

A simplified setting to mathematically study retinal waves

Bruno Cessac and Dora Karvouniari Modelling stage II retinal waves

Main hypotheses:

2) To switch from low to high state, neuron i needs an external excitatory

current. This can be:

The Ach current coming from neighbours;

A ”shot noise” current of the form:

g

N

B

i

(V

i

− V

C

)

where B ∈ {0, 1} is a Bernoulli variable with probability

Prob

[ B

i

(s) = 1, s ∈ [t, t + dt ] = pdt, with p ∼

_τ

1

.

A simplified setting to mathematically study retinal waves

Main hypotheses:

3) sAHP has a simplified form (does not depend on Calcium). We only

keep the variables S, R where S depends directly on V :

τ

S

d S

i

dt

= −S

i

+ γG (V

i

)

with:

G

(V

i

) =

1

1 + e

−

_{κ(V −V}

₎

.

A simplified setting to mathematically study retinal waves

Bruno Cessac and Dora Karvouniari Modelling stage II retinal waves

Main hypotheses:

4) Acetylcholine synapses are

assumed

not to be yet functional

(Ford-Feller, 2011). The main source of Ach coupling is Ach diffusion.

τ

A

d A

i

dt

= −A

i

+

β

A

µ

T

A

(V

i

) + D∆A

i

;

where ∆ is the Laplacian on a square lattice.

Boundary conditions are ignored.

A simplified setting to mathematically study retinal waves

1

Two state voltage.

Shot noise.

3

Simplified form for sAHP.

Ach diffusion.

Dynamics

From now we consider that variables depend continuously on space

⇒

_dt

d

→

_∂t

∂











































τ

L

∂V

_∂t

=

−(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) + ˜

g

N

B(V − V

C

)

− ˜

G

sAHP

(R)(V − V

K

) − ˜

g

A

U

(A)(V − V

A

);

τ

S

∂S

_∂t

=

−S + γG (V );

τ

R

∂R

_∂t

=

α

R

S

(1 − R) − R;

τ

A

∂A

_∂t

=

−A +

β

_µ

T

A

(V ) + D∆A(V );

Medium-scale

t

m

=

_τ

t

⇒ dt = τ

A

dt

m

ǫ

V

=

_τ

τ

∼ 5 × 10

−

3

;

ǫ

S

= ǫ

R

=

τ

_τ

∼ 4.5 × 10

−

2











































ǫ

V

_∂t

∂V

=

−(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) − g

N

B(V − V

C

)

− ˜

G

sAHP

(R)(V − V

K

) − ˜

g

A

U

(A)(V − V

A

);

∂S

∂t

=

ǫ

S

[ −S + γG (V ) ] ;

∂R

∂t

=

ǫ

R

[ α

R

S(1 − R) − R ] ;

∂A

∂t

=

−A +

β

µ

T

A

(V ) + D∆A;

Medium-scale approximation











































0

=

−(V − V

L

) − ˜

g

C

M

∞

(V )(V − V

C

) + g

N

B(V − V

C

)

− ˜

G

sAHP

(R)(V − V

K

) − ˜

g

A

U

(A)(V − V

A

);

∂S

∂t

=

0;

∂R

∂t

=

0;

∂A

∂t

=

−A +

β

µ

T

A

(V ) + D∆A;

Medium-scale approximation

At this time scale:

R, S

are constant;

V

follows the variations of R, A;

V

=

V

L

+ ˜

g

C

M

∞

(V )V

C

+ ˜

g

N

BV

C

+ ˜

G

sAHP

(R)V

K

+ ˜

g

A

U(A)V

A

1 + ˜

g

C

M

∞

(V ) + ˜

g

N

B

+ ˜

G

sAHP

(R) + ˜

g

A

U

(A)

Acetylcholine front propagation

We consider here a propagation in 1 dimension.

∂A

∂t

m

= −A +

β

A

µ

T

A

(V ) + D∆A ⇒

∂A

∂t

m

= −A +

β

A

µ

T

A

(V ) + D

∂

2

A

∂x

2

Spatially homogeneous state

Low voltage state. If all neurons are in the low voltage state, V

−

,

∆A = 0 ⇒ A reaches a stationary homogeneous state on the

medium time scale.

A

−

=

β

A

µ

T

A

(f (R, A

−

,

0)),

where 0 means that B = 0 everywhere (no shot noise current).

In this state, the S production G (V

−

) ∼ 0 so there is no sAHP on

Spatially homogeneous state

High voltage state. If all neurons are in the high voltage state,

V

+

, ∆A = 0 ⇒ A reaches a stationary homogeneous state on the

medium time scale.

A

₊

=

β

A

µ

T

A

(f (R, A

+

, B)).

Here, the state can be reached either due to shot noise or to Ach

current.

In this state, the S production G (V

+

) >> 0 so there is an sAHP

on the slow time scale.

Acetylcholine front propagation

Local perturbation of the low state. At some point x

0

we

induce a shot noise current at time 0 (B(x

0

,

0) = 1).

This raises the voltage to its upper value V

+

.

This increases the Ach production at x

0

.

Ach diffuses to neighbours and induces an excitatory current.

If g

A

conductance is large enough this current raises the

neighbours voltage to V

+

.

This can lead to a propagating wave.

Under which conditions?

Acetylcholine front propagation

Moving frame:

ξ

= x − ct

m

where c is a constant ⇒ dξ = dx − cdt

m

.

For a function A(x, t

m

) ≡ A(ξ) we have:

dA

=

∂A

∂ξ

d ξ

=

∂A

∂ξ

( dx − cdt

m

) ⇒

∂A

∂x

=

∂A

∂ξ

;

;

∂

2

A

∂x

2

=

∂

2

A

∂ξ

2

;

∂A

∂t

m

= −c

∂A

∂ξ

−c

∂A

∂ξ

= −A +

β

A

µ

T

A

(f (R, A, B)) + D

∂

2

A

∂ξ

2

Acetylcholine front propagation

Potential well:

∂V

_∂A

= A −

β

µ

T

A

(f (R, A, B))

V(A) =

A

2

2

−

β

A

µ

Z

T

A

(f (R, a, B))da + K

Rest states:

∂V

_∂A

= 0.

Newton equation

D

∂

2

_A

∂ξ

2

= −c

∂A

∂ξ

−

∂V

∂A

.

We note

∂A

_∂ξ

≡ ˙

A;

∂A

_∂ξ

≡ ¨

A

D ¨

A

= −c ˙

A

−

∂V

Acetylcholine front propagation

Bruno Cessac and Dora Karvouniari Modelling stage II retinal waves

D ¨

A

= −c ˙

A

−

∂V

∂A

.

Formally, this is the equation of motion of a particle, with mass D, in a

potential well V, with a friction coefficient c.

V has 3 extrema, given by

∂V

∂A

= A −

β

µ

T

A

(f (R, A, B)) = 0 ⇒:

A

=

β

A

µ

T

A

(f (R, A, B))

These are the fixed points of the medium scale local dynamics (∆A = 0).

Maxima of V correspond to stable fixed points.

Effect of sAHP

Figure:

Bifurcation diagram in the space I − V .

Effect of sAHP

Effect of sAHP

Figure:

Pulse propagation.

Effect of sAHP

Time and space scales

Ach raising and propagation, controlled by g

Ach

and D

(Diffusion);

sAHP controlled by g

sAHP

(Local);

Time of excitation controlled by characteristic times and

conductances;

Time of propagation controlled by c depending on the other

parameters.

Generalisations

Relaxing hypotheses

1

Two state voltage.

Shot noise.

3

Simplified form for sAHP.

Ach diffusion.

Relaxing hypotheses

1

Two state voltage. ⇒ Fast bursting in the high voltage state.

Shot noise.

3

Simplified form for sAHP.

Ach diffusion.

Bruno Cessac and Dora Karvouniari Modelling stage II retinal waves

Relaxing hypotheses

1

Two state voltage. ⇒ Fast bursting in the high voltage state.

Shot noise. ⇒ Spontaneous bursting.

3

Simplified form for sAHP.

Ach diffusion.

Relaxing hypotheses

1

Two state voltage. ⇒ Fast bursting in the high voltage state.

Shot noise. ⇒ Spontaneous bursting.

3

Simplified form for sAHP. Role of Ca ?

Ach diffusion.

Bruno Cessac and Dora Karvouniari Modelling stage II retinal waves

Relaxing hypotheses

1

Two state voltage. ⇒ Fast bursting in the high voltage state.

Shot noise. ⇒ Spontaneous bursting.

3

Simplified form for sAHP.Role of Ca ?

Ach diffusion. ⇒ Ach nicotinic coupling.

Ach nicotinic coupling

C

m

dV

i

dt

= −g

L

(V

i

− V

L

) − g

C

M

∞

(V

i

)(V

i

− V

C

) − g

K

N

i

(V

i

− V

K

)

−G

sAHP

(R

i

)(V

i

− V

K

) − g

A

(V

i

− V

A

)

X

k∈B

U

(A

k

)

Coupling two bursting cells

Karvouniari et al, ICMNS 2016

Coupling N bursting cells

Karvouniari et al, ICMNS 2016

There is a competition between 2 mechanisms:

Interburst variability which tends to desynchronise

Acetylcholine which tends to synchronise

Coupling N bursting cells

Karvouniari et al, ICMNS 2016

There is a competition between 2 mechanisms:

Interburst variability which tends to desynchronise

Acetylcholine which tends to synchronise

There is an intermediate regime of coupling, where variability

is maximum