Bursting suppression in propofol-induced general anesthesia as bi-stability in a non-linear neural mass model

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Bursting suppression in propofol-induced general anesthesia as bi-stability in a non-linear neural mass

model

Pedro Garcia-Rodriguez, Axel Hutt

To cite this version:

Pedro Garcia-Rodriguez, Axel Hutt. Bursting suppression in propofol-induced general anesthesia as bi-stability in a non-linear neural mass model. Twenty Third Annual Computational Neuroscience (CNS) meeting, Jul 2014, Quebec, Canada. �10.1186/1471-2202-15-S1-P139�. �hal-01064130�

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ERC MATHANA

Equipe NeuroSys

Tel: +33 (0)3 54 95 85 86

Mail: { pedro.garcia-rodriguez, axel.hutt } @inria.fr

INRIA CR Nancy - Grand Est. 615, rue du Jardin Botanique 54602 Villers-les-Nancy Cedex France

Bursting suppression in propofol-induced general anesthesia:

bi-stability in a non-linear neural mass model

Pedro Garc´ıa-Rodr´ıguez - Axel Hutt

SUMMARY

• Bursting activity suppression, a phenomenon characterized by sequences of alternating quiescent (’down’) and bursting (’up’) states, is ubiquitously present during deep sedation in general anesthesia and can be observed in LFPs and in EEG recordings as well [1, 2]. However, the dynamical principles underlying such phenomena are unclear.

• Prevailing theoretical approaches to this problem suggest that such bi-stable dynamics could be explained by the existence of notable non-linearities in the neuronal membrane potential as a function of the anesthetic agent, with frequent noise-induced transitions between two stable (attracting) branches.

• Nevertheless, the mathematical tractability of those models is rather limited due to the dimensionality (2D) of the phase-space.

• It has been shown for a neural-mass version of the model proposed in [3], that the 2D-trajectory likely escapes to the other attractor through a limited section of the saddle-line separating the attractions basins [4].

• This prediction is here supported by numerical simulations, permitting the simplification of the dynamics to a simpler two-state trace.

• Relevant statistics can then be extracted, including histograms of the duration of the visit in each state that may support future analytical calculations of the EEG power spectrum following renewal theory [5].

Hutt & Longtin (2009) model with no delay nor spatial dimension: a neural mass model

Differential form, using the scaled (dimensionless) time τ =√α1α2t:

∂²/∂τ²+γi∂/∂τ +ω²_i

(Vi−V_e^r) = aif(p)ω_i²Si(Ve−Vi−θi) +√ 2σW˙i

∂²/∂τ²+γe∂/∂τ + 1

(Ve−V_e^r) = aeSe(Ve−Vi−θe) +√ 2σW˙e

with γi = (β1+β2)/√α1α2, γe=p

α1/α2+p

α2/α1,ωi =p

β1β2/(α1α2). Vi, Ve are the PSPs at excitatory neurons,V_e^r the resting membrane potential,θi < θe the threshold potentials of the pre-synaptic cells, Se and Si are sigmoids functions Sk(x) = _1+e_−ck(^S^max^x_−θk), k ∈ {i, e} and ak stand for the synaptic efficacies. Wk represent Wiener processes. α1, β1, α2, β2 define the mean synaptic response functions hk:

hi(t) = aif(p) β1β2

β2 −β1

(e^−β¹^t−e^−β²^t) he(t) = ae

α₁α₂ α2−α1

(e^−α¹^t−e^−α²^t)

Finally, f(p) = r^−r/(r−1)(rp)^rp/(rp−1) mimics the inhibitory action of the propofol p level, with r=β₂/β₁, β₁ =β₀/p

Bifurcation diagrams of Hutt & Longtin (2009) model c

_i

= c

_e

(a) A triple solution case: θE > θI (b) A single solution case: θE =θI

θ

_E

= θ

_I

(a) ce < ci (b)ce=ci (c) ce > ci (d) ce >> ci

Adapted from Hutt A. and Longtin A., Cognitive Neurodyn. 4(1): 37-59, 2009.

Complexity reduction: a first-order system and a Heaviside rate function

α1 ≪α2 and β1 ≪β2 →r ≫1→f(p)≈p, leads to 1st-order Langevin (SDE) equations. Besides, replacingSk by a Heaviside function Sk= Θ(Ve−Vi−θk), the system’s equations read:

V˙i = −β(Vi−V_e^r) +βaipSmaxΘ(V₋−θi) +√ 2σW˙i

V˙e = −α(Ve−V_e^r) +αaeSmaxΘ(V−−θe) +√ 2σW˙e

whereβ =β₁ and α=α₁. The equation can also be split in three domains with a linear dynamics:

- Domain I:Ve< Vi+θi

V˙i = −β(Vi−V_e^r) +√ 2σW˙i

V˙e = −α(Ve−V_e^r) +√ 2σW˙e

- Domain II: V_e > V_i+θ_e

V˙_i = −β(V_i−V_e^r) +β₀a_iS_max+√ 2σW˙_i V˙e = −α(Ve−V_e^r) +αaeSmax+√

2σW˙e

- Domain III (V_e < V_i+θ_e and V_e> V_i+θ_i)

V˙i = −β(Vi−V_e^r) +β0aiSmax+√ 2σW˙i

V˙e = −α(Ve−V_e^r) +√ 2σW˙e

Phase portrait in Heaviside functions case: θe> θi Phase portrait in Heaviside functions case: θe=θi =θ

Heaviside variant of the model: ranges of bi-stability

Heaviside functions case: θ_e=θ_i =θ

In order to determine the ranges of bi-stability in this variant of the model, it is needed to consider the following equation:

V₋ =aeSmaxΘ(V₋−θe)−aipSmaxΘ(V₋−θi)

A first graphical analysis reveals that multiple interceptions of the l.h.s. term, the straight line f(V₋) = V₋, with the difference of the two scaled Heaviside functions on the r.h.s., are only possible in case of positive p values for (unrealistic) values of θ > 0. In such a case the V₋ coordinates of the fixed points are

V^(up)₋ = (a_e−a_ip)S_max V^(saddle)₋ = θ

V^(down)₋ = 0

and this simultaneous presence of two attractors occurs for p values bellow a critical one pc: p < pc = ^a^e_a^S_i^max_S_max^−θ. This does not exclude also unrealistic negative values for p.

Heaviside functions case: θe> θi

In this case, , for low values of p, the r.h.s. and the l.h.s. of the fixed points equation presented above always encounter in, at least, one (’up’) attractor V₋ > θ_e, located at the upper branch of the difference between the two scaled Heaviside functions (the r.h.s.):

V^(up)₋ = (ae−aip)Smax

Only this attractor is present till a new interception appears when

−aip⁽¹⁾_c Smax =θe =⇒p⁽¹⁾_c = −θe

a_iS_max

that is, for p≤p⁽¹⁾c a mono-stable (excitable) regime is observed. More interceptions will appear with increasing values of p. A second critical value p⁽²⁾c fixes the moment that the straight line f(V₋) =V₋ reaches the ’sliding’ lower branch of the Heaviside functions difference:

aeSmax−aip⁽²⁾_c Smax =θe =⇒p⁽²⁾_c = −θe+aeSmax

a_iS_max =p⁽¹⁾_c +ae

a_i

A bi-stable regime then exists for p⁽¹⁾c < p < p⁽²⁾c with two new fixed points:

V^(up)₋ = (a_e−a_ip)S_max V^(saddle)₋ = θe

V^(down)₋ = aip⁽²⁾_c Smax

The situation changes again when the moving difference reach the oblique line f(V₋) = V₋ at the lower neuronal threshold V₋ =θi:

−aip⁽³⁾_c Smax =θi =⇒p⁽³⁾_c = −θ_i aiSmax

In this case the dynamics returns to a mono-stable regime with just one quiescent state V^(down)₋ =θi for every p≥p⁽³⁾c .

Note: For the analysis above, we have assumed that p⁽²⁾c < p⁽³⁾c or, equivalently, that aeSmax <

θe−θi.

Heavise case: bifurcation diagrams

Heaviside case: θ

_e

= θ

_i

= θ Heaviside case: θ

_e

> θ

_i

Computational simulations (I)

V_dist^(up) = |V₋ − V ^(up)₋ | V_dist^(down) = |V₋ − V ^(down)₋ | V_dist^(bool) =

V_dist^(up) < V_dist^(down)

V_dist^(saddle) = V₋ − V ^(saddle)₋ V_dist^(bool) =

V_dist^(saddle) > 0

Computational simulations (II)

220 200 260 240

300 280 1.4

1.6 1.8

2 0

0.5 1 1.5

α

<T

up>

p

220 200 260 240

300 280 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0 0.5 1 1.5

α

<σ_up>

p

200 220 240 260 280 300 1.3 1.4

1.5 1.6

1.7 1.8 1.9 0

0.01 0.02 0.03

α

<T

down>

p

200 220 240 260 280 300 1.3 1.4

1.5 1.6

1.7 1.8

1.9 0.005

0.01 0.015 0.02

α

<σ_down>

p

2000 4000

6000 8000

1.3 1.4 1.5 1.6 1.7 1.8 1.9

0 0.2 0.4 0.6 0.8

σ²

<T

up>

p

2000 4000

6000 8000

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0

0.2 0.4 0.6 0.8 1

σ²

<σ_up>

p

2000 4000 6000 8000

1.3 1.4 1.5 1.6 1.7 1.8 1.9

0 0.005 0.01 0.015 0.02

σ² p

<T

down>

2000 4000 6000 8000

1.3 1.4 1.5 1.6 1.7 1.8 1.9

0 0.005 0.01 0.015 0.02

σ² p

<σ_down>

Base parameters set: θ_e =−60mV, θ_i =−70 mV, S_max= 40 Hz, V_e^r =−75mV, a_e = 0.66 mV/s, a_i = 1.2 mV/s,β₀ = 117 Hz, α= 222 Hz and σ² = 2.24∗10³

CONCLUSIONS

• The coefficient of variation Cv = _{<T >}^σ^T ≈1across the planes (p, α) and (p, σ), what might point to an intermediate-variance distribution, i.e., an exponential-like function.

• In simultaneous variations of pand α, it is found a clear decreasing trend in the mean waiting time < T(up)> for the ’up’ state whenpincreases. The contrary holds for the competing state. Conversely, none of them shows a considerable dependence on the α parameter.

• The average waiting time in the ’up’ states is a non-linear decreasing function of p, differently to the residence time in the ’down’ state that behaves mostly linearly in bothpand in the synaptic time-scale (1/α) direction.

• In simultaneous variations of pand σ, the ’up’ state is more sensitive to pchanges than toσ variations, in comparison with the behaviour observed for the ’down’ states, which ’feels’ more the variations in σ.

References

[1] Ferron J.-F. et al., J. Neuroscience 29(31): 9850-9860, 2009.

[2] Wilson M. T. et al., J Biological Physics 36: 245-259, 2010.

[3] Hutt A. and Longtin A.,Cognitive Neurodynamics 4(1): 37-59, 2009.

[4] Rodriguez P. and Hutt A., Bernstein Conference, 2013. doi: 10.12751/nncn.bc2013.0113

[5] Stratonovich R. L., Topics in the theory of random noise: Volume I, Ch. 6, Gordon and Breach, 1962.

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