Nonlinear Dynamics of Cellular and NetworkExcitability and Oscillations

CMNSS10

See review chapters: JR w/ B Ermentrout, 1998Borisyuk

w/ JR, 2005

Slow network oscillations

Case study #1: episodic rhythms in developing spinal cordCase study #2: alterations from perceptual ambiguity

Exercises #1, #2

τe

dre

/dt = -re

+ Se

(aee

re

- aei

ri

+ Ie

)

τi

dri

/dt

= -ri

+ Si

(aie

re

– aii

ri

+ Ii

)

Firing rate model (Amari-Wilson-Cowan) for dynamics of excitatory-inhibitory populations.

ri

(t), re

(t) --

average firing rate (across population and “over spikes”)

τe

, τi --

“recruitment”

time scale

Se

(input) , Si

(input) –

input/output relations

aee

etc –

“synaptic weights”

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Modeling the rhythmic dynamics of developing spinal cordJ Rinzel, NYU

Background -

network depressionAn ad hoc firing rate modelPredictions/confirmations A specific mechanism for depressionA minimal cell-based network modelMean field model? J Tabak, M O’Donovan, C Marchetti, B Vladimirski

Interneurons Retrogradely Labeled By Calcium Dyes Injected Into The Ventrolateral Funiculus (VLF) Are

Rhythmically Active (A.Ritter, P.Wenner, S.Ho)

Inject CalciumDye

LS1

Optical signals duringrhythmic activity

VLF

ventral

dorsal

rostral

caudal

Synaptic Potentials Are Transiently Depressed After An Episode Of Spontaneous Activity

200 ms

Time (minutes)

Nor

mal

ized

Am

plitu

de

epis

ode

epis

ode

τa a’ = a∞

(n.a)-a

Basic Model: mutual excitatory network.Activity level, a(t) ; availability of synaptic input, n

τa a’ = a∞

(input)-a

GABA is depolarizing

Spontaneous Rhythmic Episodes Can Be Modeled By An Excitatory Network With ‘Fast’ And ‘Slow’ Depression

J Neuroscience, 2000.

τa a’ = a∞

(s.d.a)-aτd d’ = d∞

(a)-d

Network behavior is modeled by three variables

Network ActivityActivity-dependent Fast depression

time

0.8

0.6

0.4

0.2

30 40 50

Delayed Fast Depression Leads to Oscillations

10 20Fast Depression (d)

1.0

τs s’ = s∞

(a)-sActivity-dependent Slow depression

time100 200 400 500 600

Fast and Slow Depression ProducesEpisodic Oscillations

300

Activity (a)

Slow Depression (s)

0.8

0.6

0.4

0.2

1.0

Cycling dynamics: a-d phase plane, s-fixed.

a

d

a

s

Prescribed slow back-and-forth s(t) “looks like”

episodes, but it’s not autonomous.

Framework for bursting: fast/slow analysis

HH model: repetitive firing

Example: Morris-Lecar

+ IK-Ca…slow dynamics for Ca2+(t)

Treat Ca as param, z= 1/(1+Ca)With Ca as dynamic, slow “Elliptic”

bursting

Cycling dynamics: a-d phase plane, s-fixed.

a

d

a

s

Prescribed slow back-and-forth s(t) “looks like”

episodes, but it’s not autonomous.

a

s

OSC: “cycles”

SS: silent phase

a-d phase plane, s fixed.Bistable: upper OSC, lower SS

FAST/SLOW Analysis

syn’s

depressing

syn’s

recovering

Full system: s, slow depression

“Kick”

out of Silent Phase => shorter Active Phase.

Fast/Slow Dissection --

a-s “phase plane”

THEORY EXPERIMENTJ Neuroscience, 2001.

mV

Cl

influx

Cl

efflux

V_Cl

V

FAST/SLOW Analysis

N-cell I&F network. w/ J Tabak, B Vladimirski

All-to-all excitatory coupling. Non-identical cells: I min

< I j

< I max

If I max

>1 some cells fire spontaneously.

Slow depression, no cycles.

J Computl

Neuro, 2008

sj

vs

j

gsyn

eff

raster plot

N-cell I&F network; N=50; Imin=0.1; Imax=1.1I=0 I=1

OSCEXC

Demonstration of bistability:

Use self-consistency at each t,with current values of {sj

} –find 3 states: L, M, U…

U-M coalesce at end of Silent PhaseL-M at end of Active Phase

SUMMARY: Modeling of Spontaneous Rhythms Ad hoc Mean Field model:

Episodic rhythms -

generated from mutual excitation with slow synaptic depression;

-

fast depression yields cycling during episodes.

Bistability

of fast subsystem leads to predictions forcorrelations between SP duration and next AP duration.

GABAA

synapses are “functionally”

excitatory, ie

V < VCl

;VCl

is depolarized and it oscillates together with episodes; minimal model shows it.

N-cell network model:

Cycling during AP? Stochastic synapses for initiating next AP?

Dynamics of heterogeneous network.

Slide Number 1Slide Number 2Slide Number 3Interneurons Retrogradely Labeled By Calcium Dyes Injected Into The Ventrolateral Funiculus (VLF) Are Rhythmically Active (A.Ritter, P.Wenner, S.Ho)Slide Number 5Slide Number 6Spontaneous Rhythmic Episodes Can Be Modeled By An Excitatory Network With ‘Fast’ And ‘Slow’ Depression� J Neuroscience, 2000.Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19