**4.1 The non-equilibrium potential (NEP)**

It is often hard to tell to what extent an innovation embodies a paradigm shift, for the high diversity (both in scope and extent) of innovations. The formalism of quantum mechanics can be regarded as such—with respect to the Newtonian paradigm—despite the strict correspondence between commutator and Poisson bracket Lie algebras. Also can Einstein's three papers in his "annus miraculus" be considered as such, for they demolished our former conceptions of time, of the nature of particles and waves, and of a clockwork universe. In 1908, Paul Langevin supplemented the Newtonian paradigm by letting the forces be of stochastic nature [6]. It is up to your taste to call this innovation a paradigm shift: it definitely abolished our clockwork universe conception and opened up a new chapter in the theory of differential equations. The resulting paradigm is well suited to the current situation, urged by the challenges of nanoscience (where the "systems" are submitted to strong ambient fluctuations) and favored by the increasing parallelism of computational architectures (the simulation schemes are essentially local).

to work with. A caricature of this model which nonetheless stresses its essence is thus far more desirable in many situations. The FitzHugh-Nagumo model is the minimal model capable to produce *action potentials*, and the key to this behavior is

*u*\_ ¼ *f u*ð Þ� *v,*

The *activator* field *u* relaxes very fast and displays *autocatalytic* dynamics (the more there is, the more it produces, but in a *nonlinear* fashion) as needed to produce

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>λ</sup>*1*<sup>ϵ</sup>* �<sup>1</sup> *<sup>β</sup>u*<sup>2</sup>

For *nonautonomous* cases, one can draw consequences from Eq. (20) as far as the driving is much slower than the involved relaxation times (adiabatic approxima-

The result (20) has been employed [9–13] to find the optimal noise variance *γ* for arrays of excitable elements to display *stochastic resonance synchronized behavior* (see Nomenclature). Here, we briefly illustrate one such a case, where the coupling is *inhibitory* (when neuron *i* fires, neurons *i* � 1 are less likely to fire) [14]. Inhibitory coupling is central in the dynamics of neocortical pyramidal neurons and cortical networks, and plays a major role in synchronous neural firing. On the other hand, inhibitory interneurons are more prone to couple through *gap junctions* (diffusive or "electric" coupling) than excitatory ones. In the transition from wake

We consider a ring of *N* identical excitable FHN cells, with their *inhibitor* fields

Numerical simulation of this stochastic system with increasing *γ*—for appropriate values of the diffusive coupling E between neighboring inhibitor fields—reveals the noise-induced phenomena taking place: *synchronization with the external signal* of the ring's activity and (imperfect) *spatiotemporal self-organization* of the cells. For an optimal value of *γ*, a *stochastic resonance* phenomenon takes place, and the degree of *spatiotemporal self-organization*—alternancy between two *antiphase states*

For very low *γ*, only small-amplitude and highly homogeneous [*ui*ð Þ*t* ≈ *uj*ð Þ*t* ] subthreshold oscillations (induced by the adiabatic signal) occur around the *S* ¼ 0

to anesthetic coma, for instance, diffusive coupling of inhibitor fields helps explaining the spontaneous emergence of low-frequency oscillations with spatially

*electrically* coupled to those of their nearest neighbors. The system is moreover submitted to a common *subthreshold* (see Nomenclature) harmonic signal *S t*ð Þ and *independent* additive Gaussian white noises in each component and each site, all

–2 ð*u u*0 *f x*ð Þ*dx* � �<sup>2</sup>

*:* (20)

an action potential. Its *nullcline v* ¼ *f u*ð Þ (the locus of *u*\_ ¼ 0) is a decreasing S-shaped (typically cubic) curve. On the other hand, the *inhibitor* or recovery field *v* relaxes very slowly (it mimics the time-dependent conductance of the K<sup>+</sup> channels in the axon membrane), so in the end, it enslaves the dynamics. Parameter *ϵ* is usually very large, to account for the large difference in relaxation rates. Calling *λ*<sup>1</sup> and *λ*<sup>2</sup> the eigenvalues of the diffusion tensor, the NEP for the *autonomous* system

*v*\_ ¼ *ϵ β*ð Þ *u*–*v :* (19)

*excitability*. In its minimal expression, the FHN model reads

described by Eq. (19) is [8]

*Issues in Solid-State Physics*

*DOI: http://dx.doi.org/10.5772/intechopen.84367*

Φð Þ¼ *u; v λ*<sup>2</sup>

**4.3 Arrays of excitable elements**

and temporally chaotic dynamics.

with the same variance *γ*.

(APS)—is maximum.

**13**

tion). In the following, we exploit this advantage.

�1 ð Þ *βu*–*v*

The modern approach to continuous-time dynamic flows is of *first order*. <sup>6</sup> Given an initial state *xi* of a continuous-time, dissipative, autonomous dynamic flow *x*\_ ¼ *f x*ð Þ, its conditional probability density function (PDF) *P x; t*j*xi* ð Þ *;* 0 when submitted to a (Gaussian, centered) white noise *ξ*ð Þ*t* with variance *γ*, namely,

$$\dot{\mathbf{x}} = f(\mathbf{x}) + \xi(t), \text{ with } \langle \xi(t) \rangle = \mathbf{0} \text{ and } \langle \xi(t)\xi(t') \rangle = 2\eta \delta(t - t') \tag{16}$$

obeys the Fokker-Planck equation (FPE):

$$\partial P(\mathbf{x},t|\mathbf{x},\mathbf{0}) + \partial f(\mathbf{x},t|\mathbf{x},\mathbf{0}) = \mathbf{0},\\\text{with}\\f(\mathbf{x},t|\mathbf{x},\mathbf{0}) = D^{(1)}P - \partial\_{\mathbf{x}}\left[D^{(2)}(\mathbf{x})P\right] \tag{17}$$

in terms of the "drift" *<sup>D</sup>*ð Þ<sup>1</sup> <sup>¼</sup> *f x*ð Þ and "diffusion" *<sup>D</sup>*ð Þ<sup>2</sup> <sup>¼</sup> *<sup>γ</sup>* Kramers-Moyal coefficients. Being the flow nonautonomous but dissipative, one can expect generically situations of statistical energy balance in which the PDF becomes stationary, *<sup>∂</sup>tPst*ð Þ¼ *<sup>x</sup>* 0, thus *independent of the initial state*. Then by defining the *nonequilibrium potential* <sup>Φ</sup>ð Þ *<sup>x</sup>* <sup>≔</sup> � <sup>Ð</sup> *<sup>x</sup> <sup>x</sup>*<sup>0</sup> *f y* ð Þ*dy*, it is immediate to find

$$P\_{\mathfrak{sl}}(\mathbf{x}) = \mathcal{N}(\mathbf{x}\_0) \exp\left[-\Phi(\mathbf{x})/\eta\right]. \tag{18}$$

For *n*-component dynamic flows, Φð Þ **x** is defined as � lim<sup>γ</sup>!<sup>0</sup> γ ln *Pst*ð Þ **x**; *γ* [7], but finding it ceases to be a straightforward matter.<sup>7</sup> The purpose of this section is to illustrate its *usefulness* when known. It is a *Lyapunov function* for the deterministic dynamics, and the barriers for activated processes can be straightforwardly computed lim*<sup>γ</sup>*!<sup>0</sup> *γ* ln.

#### **4.2 The FitzHugh-Nagumo model and its NEP**

Neurons communicate with each other through "action potentials," which are pulsed variations in the polarization of their membranes. The celebrated Hodgkin-Huxley model of neural physiology was one of the great scientific achievements of the past century. When the goal is insight, however, it is too cumbersome a model

<sup>6</sup> Recall it was Hamilton who first succeeded in casting conservative systems as first-order ones. In so doing, he put *coordinates* and *momenta* on the same footing. Systems are *conservative* if their phase space does not contract.

<sup>7</sup> A key is to ensure the multidimensional version of *D*ð Þ<sup>2</sup> (a symmetric tensor) to be *nonsingular*.

**4.1 The non-equilibrium potential (NEP)**

It is often hard to tell to what extent an innovation embodies a paradigm shift, for the high diversity (both in scope and extent) of innovations. The formalism of quantum mechanics can be regarded as such—with respect to the Newtonian paradigm—despite the strict correspondence between commutator and Poisson bracket Lie algebras. Also can Einstein's three papers in his "annus miraculus" be considered as such, for they demolished our former conceptions of time, of the nature of particles and waves, and of a clockwork universe. In 1908, Paul Langevin

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

supplemented the Newtonian paradigm by letting the forces be of stochastic nature [6]. It is up to your taste to call this innovation a paradigm shift: it definitely abolished our clockwork universe conception and opened up a new chapter in the theory of differential equations. The resulting paradigm is well suited to the current situation, urged by the challenges of nanoscience (where the "systems" are submitted to strong ambient fluctuations) and favored by the increasing parallelism of computational architectures (the simulation schemes are essentially local). The modern approach to continuous-time dynamic flows is of *first order*.

an initial state *xi* of a continuous-time, dissipative, autonomous dynamic flow *x*\_ ¼ *f x*ð Þ, its conditional probability density function (PDF) *P x; t*j*xi* ð Þ *;* 0 when submitted to a (Gaussian, centered) white noise *ξ*ð Þ*t* with variance *γ*, namely,

*<sup>∂</sup>P x*ð Þþ *; <sup>t</sup>*j*x;* <sup>0</sup> *<sup>∂</sup> J x*ð Þ¼ *; <sup>t</sup>*j*x;* <sup>0</sup> <sup>0</sup>*,* with *J x*ð Þ¼ *; <sup>t</sup>*j*x;* <sup>0</sup> *<sup>D</sup>*ð Þ<sup>1</sup> *<sup>P</sup>* � *<sup>∂</sup><sup>x</sup> <sup>D</sup>*ð Þ<sup>2</sup> ð Þ *<sup>x</sup> <sup>P</sup>*

*<sup>∂</sup>tPst*ð Þ¼ *<sup>x</sup>* 0, thus *independent of the initial state*. Then by defining the *non-*

dynamics, and the barriers for activated processes can be straightforwardly

in terms of the "drift" *<sup>D</sup>*ð Þ<sup>1</sup> <sup>¼</sup> *f x*ð Þ and "diffusion" *<sup>D</sup>*ð Þ<sup>2</sup> <sup>¼</sup> *<sup>γ</sup>* Kramers-Moyal coefficients. Being the flow nonautonomous but dissipative, one can expect generically situations of statistical energy balance in which the PDF becomes stationary,

For *n*-component dynamic flows, Φð Þ **x** is defined as � lim<sup>γ</sup>!<sup>0</sup> γ ln *Pst*ð Þ **x**; *γ* [7], but finding it ceases to be a straightforward matter.<sup>7</sup> The purpose of this section is to illustrate its *usefulness* when known. It is a *Lyapunov function* for the deterministic

Neurons communicate with each other through "action potentials," which are pulsed variations in the polarization of their membranes. The celebrated Hodgkin-Huxley model of neural physiology was one of the great scientific achievements of the past century. When the goal is insight, however, it is too cumbersome a model

<sup>6</sup> Recall it was Hamilton who first succeeded in casting conservative systems as first-order ones. In so doing, he put *coordinates* and *momenta* on the same footing. Systems are *conservative* if their phase space

<sup>7</sup> A key is to ensure the multidimensional version of *D*ð Þ<sup>2</sup> (a symmetric tensor) to be *nonsingular*.

*<sup>x</sup>*<sup>0</sup> *f y* ð Þ*dy*, it is immediate to find

*Pst*ð Þ¼ *x N x*ð Þ<sup>0</sup> exp ½ � �Φð Þ *x =γ :* (18)

<sup>0</sup> h i ð Þ ¼ 2*γδ t* � *t*

*x*\_ ¼ *f x*ð Þþ *ξ*ð Þ*t ,* with h i *ξ*ð Þ*t* ¼ 0 and *ξ*ð Þ*t ξ t*

obeys the Fokker-Planck equation (FPE):

**4.2 The FitzHugh-Nagumo model and its NEP**

*equilibrium potential* <sup>Φ</sup>ð Þ *<sup>x</sup>* <sup>≔</sup> � <sup>Ð</sup> *<sup>x</sup>*

computed lim*<sup>γ</sup>*!<sup>0</sup> *γ* ln.

does not contract.

**12**

<sup>6</sup> Given

(17)

<sup>0</sup> ð Þ (16)

h i

to work with. A caricature of this model which nonetheless stresses its essence is thus far more desirable in many situations. The FitzHugh-Nagumo model is the minimal model capable to produce *action potentials*, and the key to this behavior is *excitability*. In its minimal expression, the FHN model reads

$$
\dot{u} = f(u) - v,\\
$$

$$
\dot{v} = \epsilon(\beta u - v). \tag{19}
$$

The *activator* field *u* relaxes very fast and displays *autocatalytic* dynamics (the more there is, the more it produces, but in a *nonlinear* fashion) as needed to produce an action potential. Its *nullcline v* ¼ *f u*ð Þ (the locus of *u*\_ ¼ 0) is a decreasing S-shaped (typically cubic) curve. On the other hand, the *inhibitor* or recovery field *v* relaxes very slowly (it mimics the time-dependent conductance of the K<sup>+</sup> channels in the axon membrane), so in the end, it enslaves the dynamics. Parameter *ϵ* is usually very large, to account for the large difference in relaxation rates. Calling *λ*<sup>1</sup> and *λ*<sup>2</sup> the eigenvalues of the diffusion tensor, the NEP for the *autonomous* system described by Eq. (19) is [8]

$$\Phi(u,v) = \lambda\_2^{-1} (\beta u \cdot v)^2 + (\lambda\_1 e)^{-1} \left(\beta u^2 \text{-2} \int\_{u\_0}^u f(\varkappa) d\varkappa \right)^2. \tag{20}$$

For *nonautonomous* cases, one can draw consequences from Eq. (20) as far as the driving is much slower than the involved relaxation times (adiabatic approximation). In the following, we exploit this advantage.

#### **4.3 Arrays of excitable elements**

The result (20) has been employed [9–13] to find the optimal noise variance *γ* for arrays of excitable elements to display *stochastic resonance synchronized behavior* (see Nomenclature). Here, we briefly illustrate one such a case, where the coupling is *inhibitory* (when neuron *i* fires, neurons *i* � 1 are less likely to fire) [14]. Inhibitory coupling is central in the dynamics of neocortical pyramidal neurons and cortical networks, and plays a major role in synchronous neural firing. On the other hand, inhibitory interneurons are more prone to couple through *gap junctions* (diffusive or "electric" coupling) than excitatory ones. In the transition from wake to anesthetic coma, for instance, diffusive coupling of inhibitor fields helps explaining the spontaneous emergence of low-frequency oscillations with spatially and temporally chaotic dynamics.

We consider a ring of *N* identical excitable FHN cells, with their *inhibitor* fields *electrically* coupled to those of their nearest neighbors. The system is moreover submitted to a common *subthreshold* (see Nomenclature) harmonic signal *S t*ð Þ and *independent* additive Gaussian white noises in each component and each site, all with the same variance *γ*.

Numerical simulation of this stochastic system with increasing *γ*—for appropriate values of the diffusive coupling E between neighboring inhibitor fields—reveals the noise-induced phenomena taking place: *synchronization with the external signal* of the ring's activity and (imperfect) *spatiotemporal self-organization* of the cells. For an optimal value of *γ*, a *stochastic resonance* phenomenon takes place, and the degree of *spatiotemporal self-organization*—alternancy between two *antiphase states* (APS)—is maximum.

For very low *γ*, only small-amplitude and highly homogeneous [*ui*ð Þ*t* ≈ *uj*ð Þ*t* ] subthreshold oscillations (induced by the adiabatic signal) occur around the *S* ¼ 0 rest state. As *γ* increases, so does the number of cells that become noise-activated during roughly half a cycle of the external signal. For *γ* even higher, the cells' activity enhances its coherence with the external signal as a consequence of its coupling-mediated self-organization: as one neuron activates, it usually inhibits its nearest neighbors. The outcome of this phenomenon is the APS, which partially arises along the ring during the stage of activation by noise. In this scenario, noise (together with coupling and signal) plays a constructive role. Nonetheless for *γ* too large, the sync becomes eventually degraded.

### **4.4 Spatiotemporal pattern formation in arrays of FHN neurons**

We exploit the knowledge of the NEP in Eq. (20) to attempt an analytical description of the problem in Section 4.3. The case of *perfect* spatiotemporal selforganization would be equivalent to a two-neuron system with variables *u*1, *u*2, *v*1, and *v*<sup>2</sup> and PBC. This simple model allows the formation of an *antiphase* state. Since a NEP cannot be easily found for this system—and with the only purpose of calculating barrier heights—we further reduce this description by *projecting the dynamics along the corresponding slow manifolds*:

$$
\epsilon \beta \nu\_{1,2} - \nu\_{1,2} + 2E \left( \nu 2 + \nu \mathbf{1} - \nu\_{1,2} \right) = \mathbf{0}.\tag{21}
$$

dynamics of a ring of FHN cells—with nearest neighbor *electric* (diffusive) coupling between their *inhibitor* fields—undergoing spatiotemporal pattern formation induced by noise and coupling. By means of a simple model for which a NEP can be found, the mechanism whereby the process takes place was investigated

The author is deeply indebted with his coauthors G.G. Izús, A.D. Sánchez, and M.G. dell'Erba from IFIMAR-CONICET (Faculty of Exact and Natural Sciences) and D.A. Mirabella and C.M. Aldao from INTEMA-CONICET (Faculty of Engineering) of the National University of Mar del Plata (UNMdP), Argentina, with whom he undertook part of the work referred to here. Support by UNMdP, through

Ideal gas the (identical) particles composing such a gas do not interact between themselves. Free electrons they are not submitted to any external (e.g., crystal)

Chemical potential it is the cost of adding a particle to the system. For two

Stochastic resonance nonlinear systems may display the property of amplifying

IFIMAR, Faculty of Exact and Natural Sciences, National University of Mar del

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

the right intensity.

open systems (which can exchange matter and energy with their environments) to come to equilibrium, not only their temperatures but their chemical potentials must be

a *subthreshold* input signal in the presence of noise with

potential.

equal. Subthreshold unable by itself to drive a transition.

analytically.

**Acknowledgements**

*Issues in Solid-State Physics*

*DOI: http://dx.doi.org/10.5772/intechopen.84367*

**Nomenclature**

**Author details**

Roberto Raúl Deza

**15**

Plata, CONICET, Argentina

\*Address all correspondence to: deza@mdp.edu.ar

provided the original work is properly cited.

Grant EXA826–15/E779, is acknowledged.

The projected two-variable system turns out to be gradient, a situation in which a NEP can always be found. As a consequence of the PBC, the NEP landscape along the slow manifolds is symmetric with respect to the *u*<sup>1</sup> ¼ *u*<sup>2</sup> line. For *E* ¼ 0*:*5 and maximum signal amplitude, the system has two *uniform* attractors (both cells inhibited, both cells activated), two APS (with one cell activated and one inhibited) with the *same* value of Φð Þ *u*1*; u*<sup>2</sup> , four saddles, and one maximum. For *S* ¼ 0 instead, the uniform attractor with both cells activated has collapsed with the maximum, and, hence, two saddles have disappeared.

When the value of Φð Þ *u*1*; u*<sup>2</sup> at the uniform attractor, either APS and either corresponding saddle, is plotted as a function of *S*, one can see the following:


One then understands the picture: as *S* increases, whatever of the APS is chosen. As *S* decreases past the collapse, only the uniform attractor survives. However, the neuron which was activated before has not recovered completely. Hence in the next signal cycle, the other APS is more likely to appear.
