PDE models of neural networks Beno t Perthame Introduction The - - PowerPoint PPT Presentation

PDE models of neural networks

Beno ˆ ıt Perthame

Introduction The electrically active cells are characterized by an action potential

• Hodgkin-Huxley
• FitzHugh-Nagumo
• Morris-Lekar
• Izhikevich
• Mitchell-Schaeffer

Introduction

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Solutions to the Hodgkin-Huxley model and to the FitzHugh-Nagumo model

These models are accurate but very expensive/difficult to use for large assemblies of neurones.

Introduction The Wilson-Cowan model (1972) describes the firing rates N(t, x) of neuron assemblies located at position x through an integral equation d dtN(x, t) = −N(x, t) +

• w(x, y)σ
• N(y, t)
• dy + s(x, t)

Feature : multiple steady states and bifurcation theory (Bressloff-Golubitsky, Chossat-Faugeras)

• σ(·) = sigmoid
• wij = connectivity matrix
• s = source

Introduction The Wilson-Cowan model (1972) describes the firing rates N(t, x) of neuron assemblies located at position x through an integral equation d dtN(x, t) = −N(x, t) +

• w(x, y)σ
• N(y, t)
• dy + s(x, t)

Feature : multiple steady states and bifurcation theory (Bressloff-Golubitsky, Chossat-Faugeras) Aim : large scale brain activity, visual hallucinations (Kl¨ uver, Oster, Siegel...)

.

OUTLINE OF THE LECTURE I. Principle of Noisy Integrate and Fire model II. The nonlinear Noisy Integrate and Fire model

• III. The elapsed time approach

Leaky Integrate and Fire The Leaky Integrate & Fire model is simpler dV (t) =

• − V (t) + I(t)
• dt + σdW(t),

V (t) < VFiring V (t−) = VFiring = ⇒ V (t+) = Vreset. The idea was introduced by L. Lapicque (1907).

• I(t) input current
• Noise or not
• Stochastic firing

Leaky Integrate and Fire

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Solution to the LIF model

• N. Brunel, V. Hakim, W. Gerstner and W. Kistler...
• Fit to measurements
• Explains qualitatively observations on the brain activity

Leaky Integrate and Fire Written in terms of PDEs, the probability n(v, t) to find a neuron at the potential v

                    

∂n(v,t) ∂t

+ ∂

∂v leak+external currents

• − v + I(t)
• n(v, t)
• −

Noise

• a∂2n(v, t)

∂v2 =

neurons reset

• δ(v = VR)N(t),

v ≤ VF, n(VF, t) = 0, n(−∞, t) = 0, N(t) := −a∂n(VF ,t)

∂v

≥ 0, (the total flux of neurons firing at VF). N(t) is also a Lagrange multiplier for the constraint

VF

−∞ n(v, t)dv = 1.

Leaky Integrate and Fire

              

∂n(v,t) ∂t

+ ∂

∂v

• − v + I(t)
• n(v, t)
• − a∂2n(v,t)

∂v2

= δ(v = VR)N(t), v ≤ VF, n(VF, t) = 0, p(−∞, t) = 0, N(t) := −a∂n(VF ,t)

∂v

≥ 0, (the total flux of firing neurons at VF). Properties (M. C´ aceres, J. Carrillo, BP) The solutions satisfy

• n ≥ 0,

VF

−∞ n(v, t)dv = 1,

• For I(t) ≡ 0, n(v, t) −

→

t→∞ P(v) the unique steady state of integral 1

(desynchronization)

• The convergence rate is exponential

Leaky Integrate and Fire The proof uses

• the Relative Entropy

d dt

VF

−∞ P(v)H

n(v, t)

P(v)

• dv ≤ 0,

for H(·) convex,

• Hardy/Poincar´

e inequality,

VF

−∞ P(v)|u(v)|2dv ≤ C

VF

−∞ P(v)|∇u(v)|2dv,

for

VF

−∞ P(v)u(v)dv = 0 [notice P(VF) = 0].

Ledoux, Barthe and Roberto

Noisy LIF networks For networks, the current I(t) is related to the total activity of the network

            

∂n(v,t) ∂t

+ ∂

∂v

• − v+bN(t)
• n(v, t)
• − a
• N(t)

∂2n(v,t)

∂v2

= δVR(v)N(t), v ≤ VF, n(VF, t) = 0, n(−∞, t) = 0, N(t) := −a

• N(t)

∂

∂vn(VF, t) ≥ 0,

total flux of firing neurons at VF. Constitutive laws

• I(t) = bN(t)
• b = connectivity
• b > 0 for excitatory neurones
• b < 0 for inhibitory neurones
• a(N) = a0 + a1N

Noisy LIF networks Theorem (J. Carrillo, BP, D. Smets) Assume

• a = a0 > 0 and b < 0 (inhibitory)
• the initial data is bounded by a supersolution (in a certain sense)

Then,

• There are global solutions
• Uniformly bounded for all t > 0

Noisy LIF networks Theorem (M. C´ aceres, J. Carrillo, BP) Assume

• a ≥ a0 > 0 and b > 0
• the initial data is concentrated enough around v = VF.

Then,

• there are NO global weak solutions
• larger nonlinear diffusion does not help

Possible interpretation

• N(t) → ρδ(t − tBU).
• partial synchronization

Noisy LIF networks Theorem (M. C´ aceres, J. Carrillo, BP) Assume

• a ≥ a0 > 0 and b > 0
• the initial data is concentrated enough around v = VF.

Then,

• there are NO global weak solutions
• larger nonlinear diffusion does not help

Possible interpretation

• N(t) → ρδ(t − tBU).
• partial synchronization (see S. Ha)

Noisy LIF networks Numerical solution of the blow-up phenomena

probability density n(v) Total neuronal activity N(t)

Noisy LIF networks Theorem (Steady states) For

• b > 0 small enough, there is a unique steady state
• b > (VF − VR), 2ab < (VF − VR)2VR, then there are at least 2 steady

states

• b > 0 large enough, there are no steady states.

2 4 6 8 10 1 2 3 4 5 6 1 b3 b1.5 b0.5

Noisy LIF networks Similarity with a Keller-Segel type model by V. Calvez and R. Voituriez for microtubules arrangments on the membrane

            

∂n(z,t) ∂t

− ∂

∂z [µ(t)n(z, t)] − ∂2n(z,t) ∂z2

= 0, z ≥ 0,

∂ ∂zn(0, t) + µ(t)n(0, t) = 0, dµ(t) dt

= n(0, t) − µ(t)

L .

• Blow-up for large mass
• Smooth solutions for small mass (and stable steady state)

Elapsed time structured model

• K. Pakdaman, J. Champagnat, J.-F. Vibert have proposed to

structure by time rather than potential which is a possible coding of neuronal information

Elapsed time structured model

• s represents the time elapsed since the last discharge
• n(s, t) probability of finding a neuron in ’state’ s at time t
• p(s, N) ≤ 1 represents the firing rate of neurons in the ’state s’
• N(t) = activity of the network + external signaling

                  

∂n(s,t) ∂t elapsed time advances

• +∂n(s, t)

∂s +

firing neurons

• p(s, bN(t)) n(s, t) = 0,

n(s = 0, t) =

+∞

p(s, bN(t)) n(s, t)ds

• neurons reset

, This model always satisfies

+∞

n(s, t)ds = 1.

Elapsed time structured model

• s represents the time elapsed since the last discharge
• n(s, t) probability of finding a neuron in ’state’ s at time t
• p(s, N) ≤ 1 represents the firing rate of neurons in the ’state s’
• being given a total activity N

  

∂n(s,t) ∂t

+ ∂n(s,t)

∂s

+ p(s, bN(t)) n(s, t) = 0, n(s = 0, t) =

+∞

p(s, bN(t)) n(s, t)ds, N(t) := n(s = 0, t)

• b > 0

connectivity of the network

• excitatory neurons are represented by ∂p(s,N)

∂N

> 0

Elapsed time structured model

  

∂n(s,t) ∂t

+ ∂n(s,t)

∂s

+ p(s, bN(t)) n(s, t) = 0, n(s = 0, t) =

+∞

p(s, bN(t)) n(s, t)ds,

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N3 N2 N1 the function s → p(s, N) (refractory state+ fast transition)

Elapsed time structured model With synaptic integration

  

∂n(s,t) ∂t

+ ∂n(s,t)

∂s

+ p(s, X(t)) n(s, t) = 0, N(t) := n(s = 0, t) =

+∞

p(s, X(t)) n(s, t)ds, X(t) := b

t

0 N(t − u)ω(u)du.

Elapsed time structured model

  

∂n(s,t) ∂t

+ ∂n(s,t)

∂s

+ p(s, bN(t)) n(s, t) = 0,

1 bN(t) := n(s = 0, t) =

+∞

p(s, bN(t)) n(s, t)ds, Properties

• n ≥ 0,

∞

0 n(s, t)ds = 1,

• N(t) ≤ 1, n(s, t) ≤ 1,
• there is a unique solution,

Linear case For p ≡ p(s) then

• n(s, t) −

→

t→∞ P(s) the unique steady state.

Elapsed time structured model The proof goes through Generalized Relative Entropy d dt

∞

Φ(s)P(s)H

n(s, t)

P(s)

• ds ≤ 0,

for H(·) convex.

Elapsed time structured model Properties

• For small or large connectivity (b small or large) then

desynchronization still holds n(s, t) − →

t→∞ Pb(s)

• There are several periodic solutions (explicit),
• These are stable (observed numerically).

Elapsed time structured model

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Conclusion THANKS TO MY COLLABORATORS

• M. J. Carceres (U.Granada), J. A. Carrillo (U. A. Barcelona)

(for I & F ; J. Mathematical Neurosciences 2011)

• D. Smets (work in preparation)
• K. Pakdaman, D. Salort (Inst. J. Monod, U. Paris Diderot)

(for elapsed time ; Nonlinearity 2010)

Conclusion

THANK YOU ALL