Linear response for spiking neuronal networks with unbounded memory

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Submitted on 25 Aug 2020

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Linear response for spiking neuronal networks with

unbounded memory

Bruno Cessac, Rodrigo Cofré, Ignacio Ampuero

To cite this version:

Bruno Cessac, Rodrigo Cofré, Ignacio Ampuero. Linear response for spiking neuronal networks with unbounded memory. Dynamics Days Europe, Aug 2020, Nice / Online, France. �hal-02921842�

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Context Question Method Results Conclusions

Linear response for spiking neuronal networks with

unbounded memory

Bruno Cessac

Biovision Team, INRIA Sophia Antipolis,France. 25-08-2020

with

Rodrigo Cofr´e, CIMFAV, Facultad de Ingenier´ıa, Universidad de Valpara´ıso, Valpara´ıso, Chile. and

Ignacio Ampuero, Departamento de Inform´atica, Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile

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Spontaneous spiking activity

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Spontaneous spiking activity

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Spontaneous spiking activity

⇒

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Spontaneous spiking activity

⇒

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Spontaneous spiking activity

⇒

Sample averaging (non stationary)

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Spontaneous spiking activity

⇒

Sample averaging (non stationary)

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Spontaneous spiking activity

Variation in the correlation C12,15(t, t + 2)

Correlation variations are triggered by the stimulus, but they are

constrained by the network dynamics.

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Question

How to compute the spatio-temporal response of a neuronal network to a time dependent stimulus ?

Can we compute the time variation of spike-observables as a function of the stimulus and spontaneous dynamics ?

Linear response for spiking neuronal networks with unbounded memory,

Bruno Cessac, Ignacio Ampuero, Rodrigo Cofr´e,

submitted to J. Math. Neuro, arXiv:1704.05344

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Question

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Question

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Question

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From spiking neurons dynamics to linear response

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An Integrate and Fire neural network model with

unbounded memory

R.Cofr´e, B. Cessac, Chaos, Solitons and Fractals, 2013

Spikes

Voltage dynamics is time-continuous.

Spikes are time-discrete events(time resolution δ > 0).

Spike state ωk(n) ∈ 0, 1.

Spike pattern ω(n).

Spike block ωn

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An Integrate and Fire neural network model with

unbounded memory

Spikes

Voltage dynamics is time-continuous. Spikes are time-discrete events(time resolution δ > 0).

t_k(l )∈ [nδ, (n + 1)δ[ ⇒ ωk(n) = 1 Spike state ωk(n) ∈ 0, 1. Spike pattern ω(n). Spike block ωn m.

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An Integrate and Fire neural network model with

unbounded memory

Spikes

Spike block ωn

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An Integrate and Fire neural network model with

unbounded memory

Spikes

Spike block ωn

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An Integrate and Fire neural network model with

unbounded memory

Spikes

Spike block ωn

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A conductance-based Integrate and Fire model

M. Rudolph, A. Destexhe, Neural Comput. 2006, (GIF model)

Sub-threshold dynamics: Ck dVk dt = −gL,k(Vk − EL) −X j gkj(t, ω)(Vk− Ej) +Sk(t) + σBξk(t) Synapses αkj(t) = t τe − t τkj_H(t),

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: CkdVk dt = −gL,k(Vk − EL) −X j gkj(t, ω)(Vk− Ej) +Sk(t) + σBξk(t) Synapses 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.5 1 1.5 2 2.5 3 3.5 4 PSP t t=1 t=1.2 t=1.6 t=3 g(x) gkj(t) = gkj(tj) + Gkjαkj(t − tj) t > tj

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: CkdVk dt = −gL,k(Vk − EL) −X j gkj(t, ω)(Vk− Ej) +Sk(t) + σBξk(t) Synapses 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.5 1 1.5 2 2.5 3 3.5 4 PSP t g(x) gkj(t) = GkjX n≥0 αkj(t − t_j(n))

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt = −gL,k(Vk − EL) −X j gkj(t, ω)(Vk− Ej) +Sk(t) + σBξk(t) Synapses 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.5 1 1.5 2 2.5 3 3.5 4 PSP t g(x) gkj(t,ω) = GkjX n≥0 αkj(t−nδ)ωj(n)

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt = −gL,k(Vk − EL) −X j gkj(t, ω)(Vk− Ej) +Sk(t) + σBξk(t) Synapses 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.5 1 1.5 2 2.5 3 3.5 4 PSP t g(x) gkj(t,ω) = GkjX n≥0 αkj(t−nδ)ωj(n)

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Wkj def = GkjEj αkj(t, ω) =X n≥0 αkj(t−nδ)ωj(n) ik(t, ω) = gL,kEL+ X j Wkjαkj(t, ω)+Sk(t)+σBξk(t)

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e− 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e − 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e − 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du Vk(t, ω) = Vk(sp)(t, ω) + V (S) k (t, ω) + V (noise) k (t, ω)

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e− 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du Vk(t, ω) = Vk(sp)(t, ω) + V (S) k (t, ω) | {z } V_k(det)(t,ω) +V_k(noise)(t, ω)

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e − 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du Vk(t, ω) = Vk(sp)(t, ω) + V (S) k (t, ω) | {z } V_k(det)(t,ω) +V_k(noise)(t, ω) V_k(sp)(t, ω) = V_k(syn)(t, ω) + V_k(L)(t, ω),

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e − 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du Vk(t, ω) = Vk(sp)(t, ω) + V (S) k (t, ω) | {z } V_k(det)(t,ω) +V_k(noise)(t, ω) V_k(sp)(t, ω) = V_k(syn)(t, ω) + V_k(L)(t, ω), V_k(syn)(t, ω) = 1 Ck N X j =1 Wkj Z t τk(t,ω) Γk(t1, t, ω)αkj(t1, ω)dt1

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A conductance-based Integrate and Fire model

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A conductance-based Integrate and Fire model

Sub-threshold dynamics: Ck dVk dt + gk( t, ω ) Vk = ik(t, ω). Linear in V . Spike history-dependent. Dynamics integration Γk(t1, t, ω) = e − 1 Ck Rt t1∨τk (t,ω)gk( u,ω ) du Vk(t, ω) = Vk(sp)(t, ω) + V (S) k (t, ω) | {z } V_k(det)(t,ω) +V_k(noise)(t, ω) V_k(sp)(t, ω) = V_k(syn)(t, ω) + V_k(L)(t, ω), V_k(syn)(t, ω) = 1 Ck N X j =1 Wkj Z t τk(t, ω) Γk(t1, t, ω)αkj(t1, ω)dt1

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A conductance-based Integrate and Fire model

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A conductance-based Integrate and Fire model

Variable length Markov chain with unbounded memory Pn ω(n)ωn−1_−∞ ≡ Π ω(n),

θ − V_k(det)(n − 1, ω) σk(n − 1, ω)

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Markov chains and Gibbs distributions

Markov chain: P ω(n)ω_n−Dn−1 > 0 ⇒ ∃ a unique, invariant, ergodic distribution µ.

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Markov chains and Gibbs distributions

Markov chain: P ω(n)ω_n−Dn−1 > 0 ⇒ ∃ a unique, invariant, ergodic distribution µ. µ [ ω_mn ] = n Y l =m+D P h ω(l ) ω l −1 l −D i µ h ω_mm+D−1 i , ∀m < n ∈Z

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Markov chains and Gibbs distributions

Markov chain: P ω(n)ω_n−Dn−1 > 0 ⇒ ∃ a unique, invariant, ergodic distribution µ. µ [ ω_mn ] = n Y l =m+D P h ω(l ) ω l −1 l −D i µ h ω_mm+D−1 i , ∀m < n ∈Z φω_{l −D}l = log Phω(l ) ω l −1 l −D i

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Markov chains and Gibbs distributions

Markov chain: P ω(n)ω_n−Dn−1 > 0 ⇒ ∃ a unique, invariant, ergodic distribution µ. µ [ ωn_m] = exp n X l =m+D φl ω_{l −D}l µ h ωm+D−1_m i , ∀m < n ∈Z φω_{l −D}l = log Phω(l ) ω l −1 l −D i

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Markov chains and Gibbs distributions

Markov chain: P ω(n)ω_n−Dn−1 > 0 ⇒ ∃ a unique, invariant, ergodic distribution µ. µ h ωn_m| ω_mm+D−1i= exp n X l =m+D φl ω_{l −D}l , ∀m < n ∈Z φω_{l −D}l = log Phω(l ) ω l −1 l −D i

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Markov chains and Gibbs distributions

Markov chain: P ω(n)ω_n−Dn−1 > 0 ⇒ ∃ a unique, invariant, ergodic distribution µ. µ h ωn_m| ω_mm+D−1i= exp n X l =m+D φl ω_{l −D}l , ∀m < n ∈Z φω_{l −D}l = log Phω(l ) ω l −1 l −D i

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Gibbs distribution

Equilibrium stat. phys.

(Max. Entropy Principle). Non equ. stat. phys. Onsager theory.

Ergodic theory.

Markov chains - finite memory. Chains with complete

connections - infinite memory (Left Interval Specification).

P [ ω ] =_Z1e−βH{ω}

H {ω} =P

αλαXα{ω}

Xα(ω) = Product of spike events Hammersley, Clifford, 1971

O. Onicescu and G. Mihoc. CRAS Paris, 1935

R. Fernandez, G. Maillard, A. Le Ny, J.R. Chazottes, ...

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Gibbs distribution

Ergodic theory.

P [ ω ] =_Z1e−βH{ω}

H {ω} =P

αλαXα{ω}

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Gibbs distribution

Ergodic theory.

P [ ω ] =_Z1e−βH{ω}

H {ω} =P

αλαXα{ω}

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Gibbs distribution

Ergodic theory.

Markov chains - finite memory.

Chains with complete

P [ ω ] =_Z1e−βH{ω}

H {ω} =P

αλαXα{ω}

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Gibbs distribution

Ergodic theory.

P [ ω ] =_Z1e−βH{ω}

H {ω} =P

αλαXα{ω}

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Response to stimuli

Bruno Cessac Linear response for spiking neuronal networks with unbounded memory

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Response to stimuli

Bruno Cessac, Ignacio Ampuero, Rodrigo Cofr´e, submitted to J. Math. Neuro, arXiv:1704.05344

φ(n, ω) = log Pn ω(n) ω n−1 −∞ ≡ log Π ω(n), θ − V_k(sp)(t, ω) − V_k(S)(t, ω) σk(n − 1, ω) !

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Response to stimuli

φ(n, ω) = log Pn ω(n) ω n−1 −∞ ≡ log Π ω(n), θ −V_k(sp)(t, ω)−V_k(S)(t, ω) σk(n − 1, ω) !

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Response to stimuli

GIF model without stimulus ⇒stationary Gibbs distribution.

time-dependent stimulus S (t) ⇒non stationary Gibbs distribution.

How is the average of an observable f (ω, t) affected by the stimulus ?

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Response to stimuli

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Response to stimuli

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Response to stimuli

If S is weak enough: δµ [ f (t) ] = [ κf ∗ S ] ( t ) , (linear response).

κk,f( t − t1) = 1 Ck n=[ t ] X r =−∞ C(sp) " f (t, ·), H (1) k (r , ·) σk(r − 1, ·) Z r −1 τk(r −1,.) Sk(t1)Γk(t1, r − 1, ·)dt1 #

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Response to stimuli

κk,f( t − t1) = 1 Ck n=[ t ] X r = −∞ C(sp) " f (t,·), H (1) k (r ,·) σk(r − 1,·) Z r −1 τk(r − 1, .) Sk(t1)Γk(t1, r − 1,·)dt1 # History dependence.

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Response to stimuli

κk,f( t − t1) = 1 Ck n=[ t ] X r =−∞ C(sp) " f(t, ·), H (1) k (r , ·) σk(r − 1, ·) Zr −1 τk(r −1,.) Sk(t1)Γk(t1, r − 1, ·)dt1 #

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Response to stimuli

κk,f( t − t1) = 1 Ck n=[ t ] X r =−∞ C(sp)   f (t , ·), H(1)_k (r , ·) σk(r − 1, ·)   Z r −1 τk(r −1,.) Sk(t1)Γk(t1, r − 1, ·)dt1

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Response to stimuli

κk,f( t − t1) = 1 Ck n=[ t ] X r =−∞ C(sp) " f (t, ·), H (1) k (r , ·) σk(r − 1, ·) Z r −1 τk(r −1,.) Sk(t1)Γk(t1, r − 1, ·)dt1 #

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An illustrative example

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The simplest example of the gIF model is the ... discrete time leaky-integrate and fire model

Vk(n+1) = γ Vk(n)+ X

j

Wkjωj(n)+I0+Sk(t)+σBξk(n), if Vk(n) < θ

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Linear response

δµ(1)[ f (n) ] ∼ − N X k=1 D+1 X m=1 1 νk X l =0 γlK(1)_kmSk(n − m − l ) K(1)_k,m= C(sp)[ f (m, ·), ζk(0, .) ] ζk(r − 1, ω) = H(1)_k (r , ω) σk(r − 1, ω)

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Firing rates

Computations have been done using the INRIA Pranas software, B. Cessac et al, Frontiers in Neuroinformatics, Vol 11, page 49, (2017).

Figure:Linear response of f (ω, n) = ω_kc(n) for different values of stimulus amplitude A. From https://arxiv.org/abs/1704.05344

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Pairwise correlations

Figure:Correlation functions corresponding to the firing rate of the neuron kc=N2 as a function of the

neuron index k (abscissa), for different values of the time delay m. From https://arxiv.org/abs/1704.05344

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Higher order observable

Figure:Linear response of the observable f (n, ω) = ω_{kc −2}(n − 3)ω_kc(n).

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Range of validity

Figure:L2_{distance between the curves δµ}(1)_{[ f (n) ], δµ}(HC1)_{[ f (n) ] and}

the empirical curve, as a function of the stimulus amplitude A. Left panel show distance between rate curves and right panel distance between pairwise observable with delay.

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Main conclusions

The response to a time dependent stimulus is expressed in terms of correlations of the unperturbed dynamics.

Linear response is characterized by a convolution kernel depending on the observable, on the network dynamics and the afferent Gibbs distribution.

How to determine the potential (from data) ? Higher order terms in the response ?

Toward an extension of the concept of receptive-field ?

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Main conclusions

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Main conclusions

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Main conclusions

How to determine the potential (from data) ?

Higher order terms in the response ?

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Main conclusions

The potential is shaped from dynamics.

There is no ”thermodynamic principle” to constrain its shape.

Hammersley-Clifford expansion, whose first non trivial term is the ”Ising” term, contains too much terms. (Cofre, Cessac, PRE, 2014;

arXiv:1405.3621)

Which ones are relevant ? ⇒ Requires a method to reduce the ”dimensionality of the potential”

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Main conclusions

The potential is shaped from dynamics. There is no ”thermodynamic principle” to constrain its shape.

arXiv:1405.3621)

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Main conclusions

arXiv:1405.3621)

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Main conclusions

arXiv:1405.3621)

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Main conclusions

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Main conclusions

Similar to Volterra-expansion.

Higher order terms computed by D. Ruelle (Nonlinearity, 11(1):518, 1998. ) for hyperbolic dynamical systems ⇒ Complex expansion hard to handle from data.

Sigmoid are ”hard” to approximate with Taylor expansions.

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