From experiments to models R.C. Lambert Neuronal networks and - - PowerPoint PPT Presentation

From experiments to models

R.C. Lambert

Neuronal networks and physiopathological rhythms Université Pierre et Marie Curie- Neuroscience Paris Seine (NPS) CNRS UMR8246, INSERM U1130, UPMC UM119

How do I model a cortical neuron ?

cortical pyramidal neuron current clamp mode

Vm Ie Vm Ie

How do I model a cortical neuron ?

cortical pyramidal neuron current clamp mode

Vm Ie

resting Vm (-70mV) slow depolarization amplitude of the depolarization spikes decreasing frequency (adaptation)

K+

K+

A-

A-

K+

K+

A-

A-

• +

+ + + + + + +

V = EeqK+

dynamic equilibrium

• Assume channel only conducts one species of ion (e.g. K+)
• Consider energy change in transferring one mole of ions across the membrane:

electrical potential change = zFEeq chemical potential change = RT ln([ion]i/ [ion]o) Nernst equation : Eeq = - (RT / zF) ln([ion]i / [ion]o)

Resting membrane potential

• i
• i

K+

K+

A-

A-

• +

+ + + + + + +

V = EeqK+

• i

Resting membrane potential

In an electrolyte solution, the concentrations of all the ions are such that the solution is electrically neutral.

K+

K+

A-

A-

• +

+ + + + + + +

V = EeqK+

• i

Resting membrane potential

+

• 10Å

10Å

In an electrolyte solution, the concentrations of all the ions are such that the solution is electrically neutral.

K+

K+

A-

A-

• +

+ + + + + + +

V = EeqK+

• i

Cm

• i

Resting membrane potential

+

• 10Å

10Å

« A capacitor consists of two conductors separated by a non-conductive region » (Wikipedia :-))

• The capacitor behaves as a charge storage device
• The charge accumulation (Q) in the capacitor plates is not
• instantaneous. Potential across the capacitor exponentially rises

Q = C . Eeq − > Ic = dQ dt = C . dV dt

The value of the capacitance C (in F) is proportional to the nature of the dielectric (cste), its thickness (cste) and the area of the « plates » (size of the cell).

Cm = cmA

with cm (specific capacitance) = 10nF/mm2 and A in mm2 (0.01 to 0.1 mm2)

K+

K+

A-

A-

• +

+ + + + + + +

V = EeqK+

• i

EeqK+ Cm GK+

• i

Resting membrane potential

• The K+ conductance is not voltage dependent
• The K+ current = 0 when V=EeqK+

IK = GK . (V − EeqK)

Resting membrane potential

K+

Na+; Ca2+

ATP

Cl-

A T P

K

+

Na

+

Ca

2+

• other intracellular anions (non-permeant except bicarbonate) : proteins,

metabolites, phosphates, bicarbonate

• gradients maintained by pumps (Na/K-ATPase: electrogenic 3Na/2K)
• Veq near -70mV ➢ mostly K+ conductance (anion fluxes = 0)

• i

equilibrium

• i

EeqK+ Cm GK+ EeqNa+ GNa+ K+

K+ A-

• +

+ + + + + +

Na+

Na+

A-

Cm G

• i

Vinv = GNa . EeqNa + GK . EeqK GNa + GK

Resting membrane potential

Eleak = Vinv Gleak = 1 Rm

How do I model a cortical neuron ?

cortical pyramidal neuron current clamp mode

Vm Ie

resting Vm (-70mV) slow depolarization amplitude of the depolarization spikes decreasing frequency (adaptation) Cm G

• i

Gleak Eleak

Passive response to input

• i

Vm

Ic Ileak

Cm G

Gleak Eleak

0 = Ic + Ileak = Cm dVm dt + Gleak(Vm − Eleak) Cm dVm dt = dQ dt = Ic Ileak = Gleak(Vm − Eleak) At rest, Vm = cst Vm=Eleak rem : by convention, Iion >0 means output of cations. membrane current is defined as postive-outward

Passive response to input

• i

Vm

Ie Ic Ileak

Cm G

Gleak Eleak

−Ie + Ic + Ileak = − Ie + Cm dVm dt + Gleak(Vm − Eleak) = 0 Cm dVm dt = dQ dt = Ic Ileak = Gleak(Vm − Eleak) rem : by convention, current that enters the neuron through an electrode is defined as positive-inward −Ie + Ic + Ileak = 0 Cm dVm dt = Ie − Gleak(Vm − Eleak) (Kirch-hoff’s law : sum of all the currents entering a point in a circuit must be 0)

ΔVm

Rm and Cm define how the membrane potential of the cell changes in response to ion fluxes (synaptic activity,

electrode, voltage dependent conductances …) both in term of amplitude (∆Vm= Rm.Ie) and kinetics (𝛖m= Rm.Cm)

Passive response to input

Rm . Cm . dVm dt = Rm . Ie − (Vm − Eleak) Rm . Cm . dVm dt = (Vm∞ − Eleak) − (Vm − Eleak) Rm . Cm . dVm dt = Vm∞ − Vm Vm = Vm∞ . [1 − exp( −t Rm . Cm )] Vm = Vm∞ . [1 − exp( −t τm )] ; with τm = Rm . Cm (time cste) Cm dVm dt = Ie − Gleak(Vm − Eleak)

How do I model a cortical neuron ?

cortical pyramidal neuron current clamp mode

Vm Ie

resting Vm (-70mV) slow depolarization amplitude of the depolarization spikes decreasing frequency (adaptation) Cm G

• i

Gleak Eleak

The simplest neuron model : Leaky Integrate and Fire

• nly passive properties
• when Vm reaches a threshold value Vth (-50, -55mV) the neuron fires an AP and Vm is reset to Vreset (Eleak)
• if Ie=0, Vm relaxes exponentially to Eleak with time constant 𝝊

How do I model a cortical neuron ?

Cm dVm dt = Ie − Gleak(Vm − Eleak)

How do I model a cortical neuron ?

cortical pyramidal neuron current clamp mode

Vm Ie

resting Vm (-70mV) slow depolarization amplitude of the depolarization spikes decreasing frequency (adaptation)

More complex Integrate and Fire

• when Vm reaches a threshold value Vth (-50, -55mV) :
• the neuron fires an AP
• Vm is reset to Vreset (Eleak)
• Gadapt is increased by an amount ΔGadapt -> during repetitive firing Iadapt progressively

increases, slowing spike generation with Gadapt cortical neuron see for example : Gerstner, W. (2008). Spike-response model. Scholarpedia, 3(12), 1343. http://doi.org/10.4249/scholarpedia.1343

How do I model a cortical neuron ?

decreasing frequency (adaptation)

Cm dVm dt = Ie − Gleak(Vm − Eleak) − Gadapt(Vm − Eadapt)

Adding voltage-dependent conductances

EeqNa+ EeqK+ Eleak GNa+ GK+ Gleak

Hodgkin & Huxley 1939 Hodgkin & Huxley 1952

How do I model a cortical neuron ?

Cm . dVm dt = Ie − Gleak . (Vm − Eleak) − GNa(t, Vm) . (Vm − ENa) − GK(t, Vm) . (Vm − EK) Cm . dVm dt = Ie − Gleak . (Vm − Eleak) − INa(t, Vm) − IK(t, Vm)

• i

the theory

• probability of one « gate » to be open : n
• probability of the 4 « gates » to be open : n4
• maximal K+ conductance : GK
• K+ conductance of the cell : GK n4

Bezanilla 2000 Armstrong 2003

IK = GK(t, Vm) . (Vm − EK) IK = GK . n4(t, Vm) . (Vm − EK)

• probability of one « gate » to be open : n
• probability of the 4 « gates » to be open : n4
• maximal K+ conductance : GK
• K+ conductance of the cell : GK n4

1 - n(t,Vm) n(t,Vm)

• pen

probability closed probability

β(Vm) α(Vm) at a given Vm :

n(t + δt) = n(t) + α . (1 − n(t)) . δt − β . n(t) . δt n(t + δt) − n(t) δt = α . (1 − n(t)) − β . n(t) dn dt = α − (α + β) . n 1 α + β . dn dt = α α + β − n with τ = 1 α + β and n∞ = α α + β τ . dn dt = n∞ − n n(t) = n0 − (n0 − n∞) . (1 − e

−t τ )

IK = GK . n4(t, Vm) . (Vm − EK) ; the kinetics of the current at Vm is defined by τ and n∞

the theory

experiments

Recording the current evoked at different Vm :

• control Vm : voltage-clamp (impossible in vivo, non perfect in vitro, achievable for recombinant channels)
• isolate the current : pharmacology, recording solutions
• 60
• 80mV

+40

IK∞ Vm

IK = GK . n4(t, Vm) . (Vm − EK) ; the kinetics of the current at Vm is defined by τ and n∞ IK∞ = GK . n4

∞(Vm) . (Vm − EK)

n4

∞(Vm) =

IK∞ GK . (Vm − EK)

experiments

• 60
• 80mV

+40

Vm

IK = GK . n4(t, Vm) . (Vm − EK) ; the kinetics of the current at Vm is defined by τ and n∞ IK∞ = GK . n4

∞(Vm) . (Vm − EK)

n4

∞(Vm) =

IK∞ GK . (Vm − EK) = 1 1 + e

V0.5 − Vm k

n4∞

Boltzman equation

𝝊

Vm

𝝊 activation 𝝊 deactivation

experiments

IK = GK . n4(t, Vm) . (Vm − EK) ; the kinetics of the current at Vm is defined by τ and n∞

• 60
• 80mV

+40

n(t) = n0 − (n0 − n∞) . (1 − e

−t τ )

IK = GK . [n0 − (n0 − n∞) . (1 − e

−t τ )]4 . (Vm − EK)

IK = cste . (1 − e

−t τ )]4

at each fixed Vm :

EeqNa+ EeqK+ Eleak GNa+ GK+ Gleak

voltage dependent conductances have potentially 3 main states : Closed - Open - Inactivated

• 80
• 110mV
• 5

1 - m m βm(Vm) αm(Vm) 1 - h h βh(Vm) αh(Vm)

the theory

EeqNa+ EeqK+ Eleak GNa+ GK+ Gleak

voltage dependent conductances have potentially 3 main states : Closed - Open - Inactivated

m3∞ h∞

• 130mV
• 30
• 5

experiments

EeqNa+ EeqK+ Eleak GNa+ GK+ Gleak

voltage dependent conductances have potentially 3 main states : Closed - Open - Inactivated

• 130mV
• 30
• 5

𝝊(Vm)

deinactivation kinetics inactivation kinetics

experiments

Single compartment HH model

EeqNa+ EeqK+ Eleak GNa+ GK+ Gleak

Cm . dVm dt = Ie − Gleak . (Vm − Eleak) − GK . n4(t, Vm) . (Vm − EK) − GNa . m3(t, Vm) . h(t, Vm) . (Vm − ENa) … add other ionic conductances to include more complex behaviors such as adaptation

Single compartment HH model

AB/PD

PY

LP

Fast glutamatergic Slow cholinergic

b

pyloric network of the crustacean stomatogastric ganglion

• measure the properties of one voltage-dependent conductance in 10–20 neurons, a second conductance in

another 10–20 neurons, and so on … What values should be fit to describe the conductance in a model?

• fit the fastest and largest currents (measurement errors tend to make the currents smaller and slower) or fit the

mean currents

• If 2 maximal conductances are negatively correlated, then making a model that has large values of both will not

yield a realistic model. Is a neuron with mean parameters realistic ? Does it display properties shared by all of the neurons in the population ? model 1 model 2

Prinz 2004

Cellule pyramidale de la couche VI

200 µm

NRT VB

Deschênes et al. 1998

• complex geometry with « specialized » compartments : membrane

potential can take different values (dendrites, soma, axon…) -> ion fluxes within the cell tend to equalize the potentials. Intracellular medium provides a resistance to this ionic flow (highest for long and narrow branches)

How do I model a cortical neuron ?

cortical pyramidal neuron

Neuronal morphology

cellule pyramidale (cortex)

Vm can vary considerably over the surface of the celle membrane (long and narrow processes)

• complex geometry with « specialized » compartments : membrane

potential can take different values (dendrites, soma, axon…) -> ion fluxes within the cell tend to equalize the potentials. Intracellular medium provides a resistance to this ionic flow (highest for long and narrow branches)

How do I model a cortical neuron ?

Cable equation

Cm . dV dt = 1 2arl . da2 dV

dx

dx − Im + Ie

Non-linear expression that are too complex to solve analytically :

How do I model a cortical neuron ?

Cable equation

Cm . dV dt = 1 2arl . da2 dV

dx

dx − Im + Ie

synaptic currents and active conductances are ignored -> small variations around resting membrane potential, passive properties (im=(Vm-Eleak)/rm).

if a = cste ; v = Vm − Eleak

note that dv dt = dVm dt and dv dx = dVm dx

Cm . dv dt = a 2rl . d2v dx2 − v rm + Ie rm . Cm . dv dt = arm 2rl . d2v dx2 − v + rm . Ie

τ λ2

membrane time constant electrotonic length

τ . dv dt = λ2 . d2v dx2 − v + rm . Ie

How do I model a cortical neuron ?

Cable equation

constant current delta pulse constant current

How do I model a cortical neuron ?

Multi-compartment models

• Vm(x,t) for a neuron of complex morphology is much more difficult to compute
• When adding active conductances, the cable equations cannot be solved

—> Vm(x,t) must be computed numerically —> Neurons are split into separate regions (compartments) where Vm(x,t)=V(t)

cellule de Purkinje (cervelet) cellule étoilée (cortex) cellule pyramidale (cortex)

Dendrite Soma Axo

EinvSyn GSyn EeqNa+ EeqK+ GNa+ GK+ EeqNa+ EeqK+ GNa+ GK+

How do I model a cortical neuron ?

Soma Axon

EinvSyn GSyn EeqNa+ EeqK+ GNa+ GK+ EeqNa+ EeqK+ GNa+ GK+

Williams & Stuart 2002

+

Multi-compartment models How do I model a cortical neuron ?

Multi-compartment models

Stuart et al, 1997

Destexhe 1998

How do I model a cortical neuron ?

constant current injected at the periphery current injected in the soma attenuation

Multi-compartment models

Zador 1995

distance to the soma is proportional to the log of the steady-state attenuation distance to the soma is proportional to the delay

How do I model a cortical neuron ?

How do I model the cortical network ?

Mountcastle 1997

Extracellular recordings

Jian et al. 2011

Extracellular recordings

Soma Axon

EinvSyn GSyn EeqNa+ EeqK+ GNa+ GK+ EeqNa+ EeqK+ GNa+ GK+

+

Einevoll et al. 2013

Extracellular recordings

Einevoll et al. 2013

re = electrode position rn = nth compartiment position

Ơ = extracell. conductivity (cortical gray matter 0.3-0.4Sm-1)

Extracellular recordings

Einevoll et al. 2013

• cortical pyramidal neurons: synaptic input currents and return currents

separated in space. Superposition of many such open-field generators dominates cortical LFPs. (open field configuration)

• cortical stellate cells (spherically symmetrical dendritic arborization): LFPs

generated by individual synaptic inputs largely cancel out when superimposed (closed-field configuration)

Extracellular recordings

Buzsaki et al. 2012

• extracellular AP signals rapidly decrease with distance from the soma
• brief signal : time window for summation is short

Extracellular recordings

Jia et al. 2011

Jia 2011

Local Field potentials : frequency analysis

Visual cortex

• a PERIODIC function f(x) which is reasonably continuous may be expressed as the sum of a series of sine or cosine terms (called

the Fourier series), each of which has specific AMPLITUDE and PHASE coefficients known as Fourier coefficients. The raw complex values of the Fourier transforms is difficult to interpret : Power ( dominant frequencies that “match” the data) and Phase spectrum.

• Implied signal stationarity and all time resolution is lost with FT -> either split the data into small windows and calculate FT for

each window (resolution frequency > window size), or use wavelets

0-4Hz 4-8Hz 8-12Hz 12-24Hz 24-80Hz

Le Van Quyen 2007

Local Field potentials : frequency analysis

• Implied signal stationarity and all time resolution is lost with FT -> either split the data into small windows and calculate FT for

each window (resolution frequency > window size), or use wavelets

• Unlike sine and cosine waves, which are precisely localized in frequency but extend infinitely in time, wavelets are relatively

localized in both time and frequency.

Cable equation to be noted

transmembrane current Ic Ileak

Local Field potentials : CSD

• LFP measured at single location does not contain much spatial information about current entries into the cells
• Current Source Density (CSD) analysis of LFPs allows to determine the net extracellular current flow into and out of

active neuronal tissue as a function of distance Cable equation : Extracellular field potential is proportional to transmembrane currents : Current flowing in a volume (3 dimensions: x, y, z) If laminated tissue (cortex, hippocampus…) can be reduced to 1 dimension :

Local Field potentials : CSD

• LFP measured at single location does not contain much spatial information about current entries into the cells
• Current Source Density (CSD) analysis of LFPs allows to determine the net extracellular current flow into and out of active

neuronal tissue as a function of distance

𝜚A 𝜚o 𝜚B

Local Field potentials : CSD

Higley 2007

re = electrode position rn = nth compartiment position

Ơ = extracell. conductivity (cortical gray matter 0.3-0.4Sm-1) Buzsaki 2004

Spike trains recordings

Buzsaki 2004

Spike trains and Point Processes

N1 N2 N3

Functional connectivity

Mountcastle 1997 Bartho et al. 2004 sensory information

cross-correlogram

Functional connectivity

«… However, it is important to note that pair-wise correlations provide a limited view of complex population responses. … Multivariate point process models provide a full description of joint response distributions, incorporating information about receptive field properties, spiking history and interactions among cells …» Cohen & Kohn 2011

Stevenson et al. 2008

Multivariate Hawkes process

t N1 N2 N3

λ1

U Z t

Q V

=

ν

spontaneous

Generalized Linear Model (GLM)

t N1 N2 N3

λ1

U Z t

Q V

=

ν

+

h1 " 1

Τ<t

| t-Τ

R W

spontaneous self-interaction

Multivariate Hawkes process

Generalized Linear Model (GLM)

t N1 N2 N3

λ1

U Z t

Q V

=

ν

+

h1 " 1

Τ<t

| t-Τ

R W

+

hm " 1

Τ<t m!1

| t−Τ

R W

spontaneous self-interaction functional connectivity

Multivariate Hawkes process

Generalized Linear Model (GLM)

function hj→i is defined as piecewise constant on a partition of K bins of size ∂. Here, 3 preceding spikes occurring on Nj at different delays (1,2,3) will condition spike generation on Ni according to the corresponding akj→i coefficients

Lambert 2018

Multivariate Hawkes process

Generalized Linear Model (GLM)

Non-parametric adaptive estimation of functional connectivity in multivariate Hawkes models using the Least Absolute Shrinkage and Selection Operator(LASSO) method.

N1 N2 N3

λ

1

Q V τ

Q V = ν+

h1 " 1

Τ<t | t − Τ

Q V +

hm " 1

Τ<t

m!1

| t − Τ

Q V

λ

2

Q V τ

Q V = ν+

h2 " 2

Τ<t | t − Τ

Q V +

hm " 2

Τ<t

m!2

| t − Τ

Q V

λ

3

Q V τ

Q V = ν+

h3 " 3

Τ<t | t − Τ

Q V +

hm " 3

Τ<t

m!3

| t − Τ

Q V

Non-parametric adaptive estimation of functional connectivity in multivariate Hawkes models using the Least Absolute Shrinkage and Selection Operator(LASSO) method.

Multivariate Hawkes process

Generalized Linear Model (GLM)

Multivariate Hawkes process

Generalized Linear Model (GLM)

Lambert 2018

Multivariate Hawkes process

Generalized Linear Model (GLM)

Lambert 2018

50 100 150 200 250 300 350 400 detection rate (TDR, %) 50 100 150 200 250 300 350 400 excitation inhibition dataset duration (s) dataset duration (s) 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 detection rate (TDR, %) % of false excitation % of false excitation

deleting weak interaction

7 LIF networks with random excitatory and inhibitory connections

Multivariate Hawkes process

Generalized Linear Model (GLM)

Lambert 2018