Nonlinear Dynamics of Cellular and Network Excitability ...rinzel/CMNSS10/Neuronal dyns...
Transcript of Nonlinear Dynamics of Cellular and Network Excitability ...rinzel/CMNSS10/Neuronal dyns...
-
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