John Rinzel- Nonlinear Dynamics of Neuronal Systems --Cellular Level

8/3/2019 John Rinzel- Nonlinear Dynamics of Neuronal Systems --Cellular Level

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Computational NeuroscienceWhat computations are done by a neural system?

How are they done?WHAT?

Feature detectors, eg visual system.Coincidence detection for sound localization.Memory storage.Code: firing rate, spike timing.

Statistics of spike trainsInformation theoryDecision theory

Descriptive models

HOW?Molecular & biophysical mechanisms at cell &

synaptic levels firing properties, coupling.Subcircuits.System level.


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Course Schedule . * JR awayIntroduction to mechanistic and descriptive modeling, encoding concepts.Sept 6 Rinzel: Nonlinear neuronal dynamics I: mechanisms of cellular

excitability and oscillationsSept 17 Rinzel: Nonlinear neuronal dynamics II: networks.Sept 20* Simoncelli: Descriptive models of neural encoding: LNP cascadeSept 27* Paninski: Fitting LIF models to noisy spiking data

Decision-making.Oct 4 Glimcher: Neurobiology of decision making.Oct 11 Daw: Valuation and/or reinforcement learningOct 18 Rinzel: Network models (XJ Wang et al) for decision making

Vision.Oct 25 Movshon: "Cortical processing of visual motion signals"Nov 1 Rubin/Rinzel: Dynamics of perceptual bistabilityNov 8* Cai/Rangan: Large-scale model of cortical area V1.Nov 15 Tranchina: Synaptic depression: from stochastic to rate model;

application to a model of cortical suppression.Nov 22 no class (Thanksgiving)

Synchronization/correlation.Nov 29 Pesaran: Correlation between different brain areasDec 6 Reyes: Feedforward propagation in layered networks

Glimcher: Decisions, Uncertainty, and the Brain.The Science of Neuroeconomics


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Nonlinear Dynamics of Neuronal Systems-- cellular level

John RinzelComputational Modeling of Neuronal Systems

Fall 2007


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References on Nonlinear DynamicsRinzel & Ermentrout. Analysis of neural excitability and oscillations. In

Koch & Segev (see below). Also as Meth3 on www.pitt.edu/~phase/

Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometricaldissection of minimal models. In, Chow et al, eds: Models and Methods inNeurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72.

Izhikevich, EM: Dynamical Systems in Neuroscience. The Geometry of Excitability and Bursting. MIT Press, 2007.

Edelstein-Keshet, L. Mathematical Models in Biology. Random House, 1988.

Strogatz, S. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994.

References on Modeling Neuronal System Dynamics

Koch, C. Biophysics of Computation, Oxford Univ Press, 1998. Esp. Chap 7

Koch & Segev (eds): Methods in Neuronal Modeling, MIT Press, 1998.

Wilson, HR. Spikes, Decisions and Actions, Oxford Univ Press, 1999.


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Dynamics of Excitability and Oscillations

Cellular level(spiking)

Network level(firing rate)

Hodgkin-Huxley model Wilson-Cowan model

Membrane currents Activity functions

Activity dynamics in the phase plane

Response modes: Onset of repetitive activity(bifurcations)


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Nonlinear Dynamical Response Properties

Cellular: HH

brief input brief

input

Excitability Repetitive activity Bistability Bursting

Network: Wilson-Cowan(Mean field)

% f

i r i n g

Pexc

Pinhib

brief input


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Auditory brain stem neurons fire phasically, not to slowinputs. Blocking I KLT may convert to tonic.

J Neurosci, 2002


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Take Home Messages

Excitability/Oscillations : fast autocatalysis + slower negative feedback

Value of reduced models

Time scales and dynamics

Phase space geometry

Different dynamic states Bifurcations; concepts andmethods are general.

XPP software:http://www.pitt.edu/~phase/ (Bard Ermentrouts home page)


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Synaptic input many, O(10 3 to 10 4)

Dendrites, 0.1 to 1 mm long

Axon,10-10 3 mm

Classical neuron

Signals: V m ~ 100 mV membrane potentialO(msec)

ionic currents

Membrane with ion channels variable density over surface.

Dendrites graded potentials,linear in classical view

Axon characteristic impulses, propagationneurons,muscles


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Electrical Activity of Cells V = V(x,t) , distribution within cell

uniform or not?, propagation?Coupling to other cellsNonlinearitiesTime scales

V t

2 V x2Cm +Iion(V)= + I app + coupling

Current balance equation for membrane:

capacitive channels cable properties other cells

d4R i

g c,j(V j V)

g syn,j (V j(t)) (V syn -V)

Coupling: electrical - gap junctions

j

j

chemical synapsesother cells

= gk(V, W) (VV k )Iion = I ion(V, W)

kchannel types

W/ t = G(V, W)gating dynamics

generally nonlinear


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Nobel Prize, 1959


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Current Balance:(no coupling, no cable properties, steady state )

0 gK (V-V K ) + g Na (V-V Na) + g L(V-V L)

VL NaK

LL Na NaK K

gggVgVgVg

++

++

Replace E by V


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HH Recipe:

V-clamp Iion components

Predict I-clamp behavior?

IK (t) is monotonic; activation gate, nI Na(t) is transient; activation, m and

inactivation, h

e.g., g K (t) = I K (t) /(V-V K ) = G K n4(t)

with V=V clampgating kinetics:dn/dt = (V) (1-n) (V) n

= (n

(V) n)/ n(V)

n(V) increases with V.

OFF ON

P P * (V)

(V)

mass action for subunits or HH-particles

I Na(t) = G Nam3(t) h(t) (V-V Na)


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HH Equations

Cm dV/dt + G Na m3 h (V-V Na) + G K n4 (V-V K ) +G L (V-V L) = I app [+d/(4R) 2V/x2]

dm/dt = [m (V)-m]/ m(V)dh/dt = [h (V) - h]/ h(V)dn/dt = [n (V) n]/ n(V)

space-clamped

, temperaturecorrection factor = Q

10**[(temp-temp

ref )/10]

HH: Q 10=3

V

Reconstruct action potential

Time courseVelocityThresholdRefractory periodIon fluxes

Repetitive firing?


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1 m 2 has about100 Na + and K +

channels.

HH model is a mean-fieldmodel that assumes anadequate density of channels.


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Bullfrog sympatheticGanglion B cell

Cell is compact,electrically but notfor diffusion Ca 2+

MODEL:

HH circuit+ [Ca 2+] int+ [K+] ext

g c & gAHP depend on [Ca 2+] int

Yamada, Koch, Adams 89


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Cortical Pyramidal Neuron

Complex dendritic branchingNonuniformly distributed channels

Pyramidal Neuron with axonal tree

Schwark & Jones, 89

Koch, Douglas,Wehmeier 90

HH action potential biophysical time scales.


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p p y

I-V relations: ISS(V) I inst(V)steady state instantaneous

HH: I SS(V) = G Na m3(V) h (V) (V-V Na) + G K n4(V) (V-V K ) +G L (V-V L)

Iinst(V) = G Na m3(V) h (V-V Na) + G K n (V-V K ) +G L (V-V L)

fast slow, fixed at holding valuese.g., rest


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Dissection of HH Action Potential

Fast/Slow Analysis - based on time scale differences

h, n are slow relativeto V,m

Idealize AP to 4 phases V

th,n constant during

upstroke and downstroke

V,m slaved during plateauand recovery

h,n constant duringupstroke and downstroke

Upstroke

R and E stable

T - unstable

C dV/dt = - I inst(V, m (V), h R , nR ) + I app

R T E


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Repetitive Firing, HH model and others

Response to current step

nerve block

Subthreshold


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Repetitive firing in HH and squid axon-- bistability near onset

Linear stability: eigenvalues of 4x4 matrix. For reduced model

w/ m=m (V): stability if Iinst /V + C m/ n > 0.

Rinzel & Miller, 80

HH eqns Squid axon

Guttman, Lewis & Rinzel, 80

Interval of bistability

Bistability in motoneurons


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Bistability in motoneurons


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2-compartment model; plateau generator in dendrite

Morris-Lecar membrane model

Booth & Rinzel, 95


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Bistability in 2-compt model

Response to up/down ramp; hysteresis

Switching between states

bl d l h l l


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Two-variable Model Phase Plane Analysis

ICa

fast, non-inactivatingIK -- delayed rectifier, like HHs I K

Morris & Lecar, 81 barnacle musclel

VVK VL VCaVrest

negative feedback: slow


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h ll l


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Get the NullclinesdV/dt = - I

inst(V,w) + I

app

dw/dt = [ w(V) w] / w(V)

V

I inst

w = w rest

w > w rest

dV/dt = 0Iinst (V,w) = I app

w

wrest

V-nullcline

V

w-nullcline

rest state

w= w (V)

dw/dt = 0


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Case of small

traj hugs V-nullcline -except for up/down

jumps.

ML model- excitableregime


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Repetitive Activity in ML (& HH)


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Onset is viaHopf bifurcation

Type II onsetHodgkin 48

Anode Break Excitation or


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Anode Break Excitation or Post-Inhibtory Rebound (PIR)

IK - deactivated

Post inhibitory facilitation PIF


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Subthreshold nonlinearities:ipsp can enhance epsp,and lead to spiking

Post inhibitory facilitation, PIF,transient form of PIR

Model of coincidence-detectingcell in auditory brain stem.Has a subthreshold K + current I KLT .

Theory PIF Experiment (Gerbil MSO, slice)


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Competing factors:hyperpolarzn (farther from V th)and hyperexcitable (reduced w)

Dependence on inh

B ti g S t R t ith F t I hibiti i PIF


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Boosting Spontaneous Rate with Fast Inhibition, via PIF

Firing Response to Poisson-G ex train Enhanced by Inhibition (Poisson-G inh)Gex only: Add G inh: # of spikes: 10, 15

Std HH model; 100 Hz inputs; ex= inh=1 ms

w/ R Dodla. PhysRev E, in press

Adjust params changes nullclines: case of 3 rest states


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Adjust param s changes nullclines: case of 3 rest states

Stable or Unstable?

3 states not necessarily:stable unstable stable.

3 states Issis N-shaped

small enough,then both upper/middleunstable if on middle branch.

ML: large 2 stable steady states


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Neuron is bistable: plateau behavior.

Saddle point, withstable and unstable manifolds

V

t e.g., HH withVK = 24 mV

Iapp switching pulses

ML: small both upper states are unstable


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Neuron is excitable with strict threshold .

thresholdseparatrix long

Latency

Vrest

saddle

IK-A can give long latencybut not necessary.

Iss must be N-shaped.

Onset of Repetitive Firing 3 rest states


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p g

SNIC- saddle-node on invariant circle

V

wIapp

excitable

saddle-node

limit cycle

homoclinic orbit;infinite period

emerge w/ large amplitude zero frequency

Response/Bifurcation diagramML: small


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ML: small

freq ~ II1

Firing frequency starts at 0.low freq but no conductancesvery slow

IK-A ? (Connor et al 77)

Type I onsetHodgkin 48


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Wilson-Cowan Model


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dynamics in the phase plane.

Je

Phase plane, nullclines for range of J e.

Oscillator death


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- Je

Pex

Regime of repetitive activity

Subthreshold

Oscillator Death but cells are firing


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Frequency

- Je

Type I

Type II

Transition from Excitable to Oscillatory


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Type II, min freq 0Iss monotonicsubthreshold oscillnsexcitable w/o distinct thresholdexcitable w/ finite latency

Same for W-C network models.

Type I, min freq = 0ISS N-shaped 3 steady statesw/o subthreshold oscillationsexcitable w/ all or none (saddle) thresholdexcitable w/ infinite latency

Hodgkin 48 2 classes of repetiitive firing;Also - Class I less regular ISI near threshold


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Noise smooths the f I relation


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Noise smooths the f-I relation

Type II

Type I

I app

frequency


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FS cellnear threshold

RS cell, w/ noise FS cell, w/ noise

Take Home Message


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Take Home Message

Excitability/Oscillations : fast autocatalysis + slower negative feedback

Value of reduced models

Time scales and dynamics

Phase space geometry

Different dynamic states Bifurcations

XPP software:http://www.pitt.edu/~phase/ (Bard Ermentrouts home page)

John Rinzel- Nonlinear Dynamics of Neuronal Systems --Cellular Level

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