Nikolai Bessonov

Anne Beuter

Sergei Trofimchuk

Vitaly Volpert

Cortical waves and post-stroke brain stimulation

Keywords: brain stimulation, integro-differential equation, neural field model, waves

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Over the last few years, there has been an increasing interest in the application of noninvasive stimulation such as Repetitive transcranial magnetic stimulation (TMS), Transcranial random noise stimulation (tRNS), Transcranial alternating current stimulation (tACS), Transcranial direct current stimulation (tDCS) (see Vosskuhl et al [START_REF] Vosskuhl | Non-invasive brain stimulation: a paradigm shift in understanding brain oscillations[END_REF] ), or minimally invasive brain stimulation such as cortical electrical stimulation [START_REF] Canavero | Textbook of Therapeutic Cortical Stimulation[END_REF] to treat a variety of neurologic and psychiatric conditions. This growing interest is probably due in part to an increasing understanding of the physiological control mechanisms driving normal and abnormal structural and functional brain reorganization. Specifically, these mechanisms are actively explored for stroke [START_REF] Dalise | Adaptive plasticity and recovery in preclinical models of stroke[END_REF][START_REF] Kiran | What is the nature of poststroke language recovery and reorganization?[END_REF][START_REF] Takeuchi | Combinations of stroke neurorehabilitation to facilitate motor recovery: perspectives on Hebbian plasticity and homeostatic metaplasticity[END_REF] and for post stroke aphasia, [START_REF] Balossier | Epidural cortical stimulation as a treatment for poststroke aphasia: a systematic review of the literature and underlying neurophysiological mechanisms[END_REF][START_REF] Norise | Non-invasive brain stimulation in the treatment of post-stroke and neurodegenerative aphasia: parallels, differences, and lessons learned[END_REF] to name a few. Recent studies on detailed mapping of the healthy and diseased brain connectivity shed new light on the relation between brain structure and function. [START_REF] Sporns | The human connectome: origins and challenges[END_REF] Today, brain con-nectivity and brain dynamics are combined in an integrated framework. [START_REF] Kopell | Beyond the connectome: the dynome[END_REF] An interesting observation regarding brain dynamics is that oscillations at multiple frequencies form spatially continuous neural patterns such as travelling waves. They are defined as a spatially coherent oscillation that propagates progressively across the cortex with known amplitude, frequency, and speed. [START_REF] Zhang | Theta and alpha oscillations are traveling waves in the human neocortex[END_REF] These travelling waves are functionally important for human behaviors such as perception, action, and language. [START_REF] Rapela | Rhythmic production of consonant-vowel syllables synchronizes traveling waves in speech-processing brain regions arXiv[END_REF][START_REF] Rapela | Traveling waves appear and disappear in unison with produced speech[END_REF] Propagation of alpha waves in the neocortex is studied in Bahramisharif et al [START_REF] Bahramisharif | Propagating neocortical gamma bursts are coordinated by traveling alpha waves[END_REF] and Patten et al. [START_REF] Patten | Human cortical traveling waves: d properties and correlations with responses[END_REF] Travelling waves in the visual cortex are investigated in Sato et al [START_REF] Sato | Traveling waves in visual cortex[END_REF] and in motor cortex in Takahashi et al. [START_REF] Takahashi | Large-scale spatiotemporal spike patterning consistent with wave propagation in motor cortex[END_REF] In the presence of brain disorders, these waves become disorganized and brain networks become disconnected or deactivated. But controlling dynamics in neural systems via brain stimulation techniques remains challenging because as indicated by Tang and Bassett, [START_REF] Tang | Control of dynamics in brain networks[END_REF] "While some understanding has been gained of the control of single neurons, the control of large scale neural systems networks of multiply interacting components remains poorly understood." As a result, in most cortical stimulation protocols, be they electric or magnetic, parameter values are still adjusted in an empirical fashion. Despite this limitation, cortical stimulation studies clearly produce encouraging results suggesting that these techniques have some potential, but overall, the results remain mixed or limited for example for stroke (see previous studies [START_REF] Azad | Neurorestoration after stroke[END_REF][START_REF] Hatem | Rehabilitation of motor function after stroke: a multiple systematic review focused on techniques to stimulate upper extremity recovery[END_REF][START_REF] Levy | Epidural electrical stimulation for stroke rehabilitation: results of the prospective, multicenter, randomized, single-blinded everest trial[END_REF][START_REF] Lopresti | The promising role of neuromodulation in improving ischemic stroke recovery[END_REF][START_REF] Plow | Invasive cortical stimulation to promote recovery of function after stroke: a critical appraisal[END_REF][START_REF] Wessel | Non-invasive brain stimulation: an interventional tool for enhancing behavioral training after stroke[END_REF] ), with improvements sometimes limited [START_REF] Kubis | Non-invasive brain stimulation to enhance post-stroke recovery[END_REF] to 10% to 20%, and for post-stroke aphasia (see other works [START_REF] Cherney | Epidural cortical stimulation as adjunctive treatment for nonfluent aphasia: Phase 1 clinical trial follow-up findings[END_REF][START_REF] Meinzer | Electrical stimulation of the motor cortex enhances treatment outcome in post-stroke aphasia[END_REF][START_REF] Mohr | Neuroplasticity and functional recovery after intensive language therapy in chronic post stroke aphasia: which factors are relevant?[END_REF][START_REF] Santos | Non-invasive brain stimulation and computational models in post-stroke aphasic patients: single session of transcranial magnetic stimulation and transcranial direct current stimulation. A randomized clinical trial[END_REF][START_REF] Shah | Induction of neuroplasticity and recovery in post-stroke aphasia by non-invasive brain stimulation[END_REF][START_REF] Spelmann | Comparison of two configurations of transcranial direct current stimulation for treatment of aphasia[END_REF] ). Aphasia is a complex brain disorder of language and communication caused by damage to the language networks affecting speaking, reading, writing, or understanding language.

In order to propose more efficient stimulation protocols in the future, it is important to predict the dynamics of the propagating waves, to detect deviant values of propagating wave parameters (ie, wave speed, amplitude, and frequency) and to identify the precise localization of the nodes to be stimulated. These constraints imply the use of bio-inspired modeling tools to reconstruct the information content of the propagating wave and to monitor in time and space the stimulation to be introduced via a closed-loop control system to compensate for the disruption. Until now, this type of cortical stimulation was not available.

However, things may change soon for two reasons. The first reason is that the focus of research has progressively shifted from examining specific brain regions to exploring specific brain networks. [START_REF] Mandonnet | Understanding entangled cerebral networks: a prerequisite for restoring brain function with brain-computer interfaces[END_REF][START_REF] To | Changing brain networks through non-invasive neuromodulation[END_REF] The challenge to treat brain networks is to understand and control the brain's capacity to reorganize itself. [START_REF] Norise | Non-invasive brain stimulation in the treatment of post-stroke and neurodegenerative aphasia: parallels, differences, and lessons learned[END_REF] The second reason is that it is now possible to describe cortical waves mathematically with increasing precision. [START_REF] Zhang | Theta and alpha oscillations are traveling waves in the human neocortex[END_REF][START_REF] Bessonov | Estimate of the travelling wave speed for an integro-differential equation[END_REF][START_REF] Beuter | Modeling Of post-stroke stimulation of cortical tissue[END_REF] These recent modeling studies are opening the possibility to harness the brain's capacity to reorganize itself with the help of an outside stimulation introduced only when and where necessary. Thus, it may soon become possible to restore or repair interacting brain networks with the consequence of improving significantly symptoms associated with a variety of brain disorders affecting most components of human behaviors (action, emotion, and cognition).

By definition, every model is a simplification of reality, but an effort is made to render the model realistic neurophysiologically. For example, we incorporate the effect of the extracellular field on the transmembrane potential of neurons (ie, ephaptic coupling, Appendix A). In addition, the model distinguishes activation and inhibition response functions, it differentiates short range activation and long range inhibition, its connectivity function is asymmetric.

In the present paper, we study the propagation of waves of electric potential in the cortical tissue with integro-differential equations arising in neural field models. The main motivation of this work is to study the propagation of brain waves in the normal and in the damaged tissue and to show that external stimulation can restore the properties of the waves altered by the tissue damage. We will consider different stimulation techniques and will discuss the conditions of their applicability. Stability analysis presented in Section 3 will allow us to determine conditions of appearance of stationary spatial structures and of periodic travelling waves. Different regimes of wave propagations are illustrated in Section 4. The question of the response to electric stimulation for different stimulation types is addressed in Section 5. We conclude this work by a discussion of the model and of the results and by future perspectives.

MODEL

We consider the equation for the local density u(x, t) of activated (firing) neurons as a function of space point x and time t:

𝜕u 𝜕t = DΔu + g(K -u)H(W a )-g(u)H(W i )-𝜎u. (2.1)
The first term in the right-hand side of this equation describes neuron activation due to ion diffusion through the extracellular matrix, direct gap junction communication, and ephaptic effects [START_REF] Buzsaki | The origin of extracellular fields and currents EEG, ECoG, LFP and spikes[END_REF] (see Section 6). On the other hand, electrodynamic effects (second time derivative) are not taken into account. The second and third terms take into account neuron activa-tion and deactivation by signals coming from other neurons along axons. Instead of activating and inhibitory neurons in Wilson and Cowan, [START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF] we consider here a single group of neurons with activating W a and inhibitory W i signals. This group of neurons can be interpreted as ensemble of activating and inhibitory neurons sending different types of signals. The last term describes intrinsic deactivation rate, that is, not related to the inhibition signals.

We denote by K the total neuron density, so that Ku is the density of inactivated neurons. In the linear approximation, the product (Ku)W a determines the rate of neuron activation by the activating signal W a .Inamoregeneralsetting,we take into account saturation with respect to the neuron density and the force of activation signal by the introduction of some growing non-negative functions g(u) and H(W).Thus,g(Ku)H(W a ) is the rate of neuron activation and g(u)H(W i ) is the rate of their deactivation.

The activating signal is the electric potential produced by other neurons and coming to the given space point x along the axons. It can be considered in the form [START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease[END_REF] :

W a (x, t)= ∫ Ω 𝜙 a (x,𝑦)S a ( u ( 𝑦, t - |x -𝑦| q a
)) d𝑦.

(2.2)

This expression describes the intensity of signal coming from all points y in the spatial domain Ω to the point x. Here, S a (u) is the excitation generation (response function) produced by activated neurons, q a is the excitation speed, |x -y|∕q a is the time delay due to the excitation propagation from the point y to the point x,and𝜙 a (x, y) is the connectivity function.

The inhibitory signal has a similar form, Connectivity functions 𝜙 a (x, y) and 𝜙 i (x, y) characterize the density and the length of axons connecting the neurons. Let us note that signals in axons propagate only in the direction of their terminals. Therefore, signal propagation is not in general symmetric: it can be different from y to x and from x to y (see Section 6). It depends on the number and on the direction of connections between the two points. Connectivity functions are often considered as exponential functions of the distance r = xy [START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease[END_REF][START_REF] Pinotsis | Anatomical connectivity and the resting state activity of large cortical networks[END_REF] :

W i (x, t)= ∫ Ω 𝜙 i (x,𝑦)S i ( u ( 𝑦, t - |x -𝑦| q i )) d𝑦 (2.
𝜙 a (r)= { a 1 e -b 1 r , r > 0 a 3 e b 3 r , r < 0 ,𝜙 i (x)= { a 2 e -b 2 r , r > 0 a 4 e b 4 r , r < 0 , ( 2.4)
where a i , b i are some positive constants. In this work, we use the functions 𝜙 a and 𝜙 i defined in (2.4) unless otherwise is stated.

Response functions S a (u) and S i (u) are non-negative nondecreasing functions usually considered as sigmoid functions, that is, they have an interval of rapid growth followed by saturation (Figure A1).Their exact form will be specified below.

If both densities u and Ku are sufficiently large, then the function g is close to its saturation value. On the other hand, if the signals W a and W i are sufficiently small, then the function H(W) can be approximated by a linear function. In this case, we obtain the equation

𝜕u 𝜕t = DΔu + W a (x, t)-W i (x, t)-𝜎u, ( 2.5)
where the constant factors are omitted for simplicity of notation. The variable u in Equation 2.5 can be interpreted as the electric potential and not the density of activated neurons. Although these variables are related to each other and, in the first approximation, they can be considered as linearly correlated, there is a difference between them which should be taken into account. Clearly, the density of neurons cannot become negative, while the electric potential can have variable sign. In both cases, we can interpret the variable u as a deviation from an equilibrium neuron density or electric potential in the linearized model [START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease[END_REF] in such a way that this variable can change sign.

Various particular cases of this equation were considered in the literature. The case without diffusion (D = 0) and without inhibition term (𝜙 i (x, y) ≡ 0) was considered in Ermentrout and McLeod 39 as a model of neural network. A slightly modified equation was considered to describe thalamic waves. [START_REF] Chen | Wave propagation mediated by GABAB synapse and rebound excitation in an inhibitory network: a reduced model approach[END_REF] A similar equation with both activation and inhibition connectivity functions was considered in Modolo et al [START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease[END_REF] to model brain stimulation in the case of Parkinson's disease. The case with diffusion but without inhibition was studied in Bessonov et al [START_REF] Bessonov | Estimate of the travelling wave speed for an integro-differential equation[END_REF] and Beuter et al. [START_REF] Beuter | Modeling Of post-stroke stimulation of cortical tissue[END_REF] In the remaining part of the work, we will study Equation 2.5 in the limit of large excitation speed q where the time delay |x -y|∕q disappears. The case with time delay will be studied in the subsequent works.

STABILITY ANALYSIS

In the limiting case of large excitation speed, q =∞ ,andinthe1Dcase(Ω=R), Equation 2.5 writes:

𝜕u 𝜕t = D 𝜕 2 u 𝜕x 2 + ∫ ∞ -∞ (𝜙 a (x -𝑦)S a (u(𝑦, t)) -𝜙 i (x -𝑦)S i (u(𝑦, t))) d𝑦 -𝜎u. (3.1)
Homogeneous in space stationary solution u = u 0 of this equation is a solution of the equation

F(u) ≡ 𝜙 0 a S a (u)-𝜙 0 i S i (u)-𝜎u = 0, (3.2)
where

𝜙 0 a = ∫ ∞ -∞ 𝜙 a (x)dx,𝜙 0 i = ∫ ∞ -∞ 𝜙 i (x)dx.
Linearizing Equation 3.1 about u 0 and applying the Fourier transform, we get the expression for the spectrum:

𝜆(𝜉)=-D𝜉 2 + s a φa (𝜉)-s i φi (𝜉)-𝜎, (3.3)
where s a = S ′ a (u 0 ), s i = S ′ i (u 0 ), φa (𝜉) and φi (𝜉) are the Fourier transforms of the corresponding functions. The variable 𝜉 adopts all real values if we consider the whole axis and some discrete set of values for a bounded interval (say, [0, L])with the periodic boundary conditions.

Suppose that 𝜆(0)=F ′ (u 0 ) < 0, where the function F(u) is defined in (3.2). Then in the narrow connectivity limit, where the functions 𝜙 a and 𝜙 i are replaced by the 𝛿-function and the integral term in Equation 3.1 is replaced by 𝜙 a S a (u),𝜙 i S i (u), the homogeneous in space stationary solution u = u 0 is asymptotically stable. We will verify whether 𝜆(𝜉) can have a positive real part for other real values of 𝜉 providing instability of the solution u 0 .Wehave

φa (𝜉)= ∫ ∞ -∞ 𝜙 a (x)e ix𝜉 dx = ∫ ∞ -∞ 𝜙 a (x) cos(𝜉x)dx + i ∫ ∞ -∞ 𝜙 a (x) sin(𝜉x)dx = a 1 b 1 b 2 1 + 𝜉 2 + a 3 b 3 b 2 3 + 𝜉 2 + i𝜉 ( a 1 b 2 1 + 𝜉 2 - a 3 b 2 3 + 𝜉 2 ) , φi (𝜉)= a 2 b 2 b 2 2 + 𝜉 2 + a 4 b 4 b 2 4 + 𝜉 2 + i𝜉 ( a 2 b 2 2 + 𝜉 2 - a 4 b 2 4 + 𝜉 2
) .

Activation without inhibition

If 𝜙 i (x) ≡ 0, then Re 𝜆(𝜉) ≤ 𝜆(0) < 0forallreal𝜉. Therefore, the homogeneous in space solution u 0 is stable.

Symmetric connectivity functions

Suppose that

a 1 = a 3 , b 1 = b 3 , a 2 = a 4 , b 2 = b 4 . (3.4)
Then,

φa (𝜉)= 2a 1 b 1 b 2 1 + 𝜉 2 , φi (𝜉)= 2a 2 b 2 b 2 2 + 𝜉 2 .
Suppose that s a φa (0) < s i φi (0) and s a φa (𝜉) > s i φi (𝜉) for some 𝜉.

(3.5)

The first condition is satisfied if s a a 1 b 2 < s i a 2 b 1 , and it implies that 𝜆(0) < 0. The second condition is satisfied if

(s a a 1 b 1 -s i a 2 b 2 )𝜉 2 > b 1 b 2 (s i a 2 b 1 -s a a 1 b 2 ).
Since the right-hand side of this inequality is positive, then it can be satisfied for some 𝜉 if and only if

s a a 1 b 1 > s i a 2 b 2 .
We can now formulate the following assertion.

Proposition 3.1. Suppose that the following inequalities are satisfied:

b 1 b 2 > s a a 1 s i a 2 , b 1 b 2 > s i a 2 s a a 1 . (3.6)
Then properties (3.5) hold.

It follows from conditions (3.6) that b 1 > b 2 . Hence, the activation connectivity function decays faster than inhibition connectivity function. This condition can be formulated as short range activation long range inhibition, similar to the case of Turing instability. Under the conditions of Proposition 3.1, solution u 0 becomes unstable for D and 𝜎 sufficiently small.

We recall that stability boundary is determined by the values of parameters for which the maximal real part of the spectrum lies at the imaginary axis. Setting 𝜆(𝜉)=0 in (3.3), we get 𝜎 as a function of 𝜉 at the stability boundary:

𝜎 = 2s a a 1 b 1 b 2 1 + 𝜉 2 - 2s i a 2 b 2 b 2 2 + 𝜉 2 -D𝜉 2 ≡ Φ 1 (𝜉). (3.7)
Figure 1 (left) shows two examples of the stability boundary. The upper curve has positive values in some interval of frequencies 𝜉. For the values of 𝜎 less than the maximum of this function, the maximal eigenvalue is positive leading to the emergence of a periodic in space solution (Figure 1, right). The lower curve is nonpositive. Therefore, the emergence of periodic solutions cannot be observed. Since the maximum of this function equals zero, it separates the cases with and without instability.

Let us recall that in the case of a bounded interval, we have a discrete set of values 𝜉 j = 2𝜋j∕L, j = 1, 2, .... The instability occurs if 𝜎<Φ 1 (𝜉 j ) for some j. If this condition is satisfied for more than one value of j, then solutions with different frequencies can bifurcate.

The maximum of the bifurcating solution as a function of 𝜎 is shown in the bifurcation diagram (Figure 2). We observe a typical square root dependence near the bifurcation point. It is interesting to note that amplitude growth accelerates for further decrease of 𝜎.

Asymmetric connectivity functions

Suppose now that conditions (3.4) are not satisfied. In this case, the eigenvalue (3.3) has a nonzero imaginary part. Set 𝜆(𝜉)=𝛼(𝜉)+i𝛽(𝜉),where

𝛼(𝜉)= s a a 1 b 1 b 2 1 + 𝜉 2 + s a a 3 b 3 b 2 3 + 𝜉 2 - s i a 2 b 2 b 2 2 + 𝜉 2 - s i a 4 b 4 b 2 4 + 𝜉 2 -D𝜉 2 -𝜎, 𝛽(𝜉)=s a 𝜉 ( a 1 b 2 1 + 𝜉 2 - a 3 b 2 3 + 𝜉 2 ) -s i 𝜉 ( a 2 b 2 2 + 𝜉 2 - a 4 b 2 4 + 𝜉 2 ) . (3.8)
As before, the stability boundary is given by the condition 𝛼(𝜉)=0: Here, 𝜉 is found from the equation 𝛼(𝜉)=0and𝛽 is given by equality (3.8). This solution represents a periodic wave with frequency 𝜉 and speed c =-𝛽∕𝜉. We get from (3.8):

𝜎 = s a a 1 b 1 b 2 1 + 𝜉 2 + s a a 3 b 3 b 2 3 + 𝜉 2 - s i a 2 b 2 b 2 2 + 𝜉 2 - s i a 4 b 4 b 2 4 + 𝜉 2 -D𝜉 2 ≡ Φ 2 (𝜉). ( 3
c =-s a ( a 1 b 2 1 + 𝜉 2 - a 3 b 2 3 + 𝜉 2 ) + s i ( a 2 b 2 2 + 𝜉 2 - a 4 b 2 4 + 𝜉 2
) .

(3.10)

If there are two (or more) different frequencies 𝜉 1 and 𝜉 2 satisfying the condition 𝜎<Φ(𝜉 j ), j = 1, 2, then there are two waves with different speeds and frequencies. According to (3.10), the value of the speed can increase with frequency or decrease depending on the values of parameters.

REGIMES OF WAVE PROPAGATION

Monotone waves

Replacing the connectivity functions 𝜙 a and 𝜙 i by the 𝛿-function, we obtain conventional reaction-diffusion equation

𝜕u 𝜕t = D 𝜕 2 u 𝜕x 2 + S a (u)-S i (u)-𝜎u. (4.1)
For the typical functions S a (u) and S i (u) (Figure A1), the function f(u)=S a (u)-S i (u)-𝜎u has three non-negative zeros, u + , u 0 , u -, u + < u 0 < u -if 𝜎 is less than some critical value 𝜎 c . In this case, there is a travelling wave solution of this equation, that is, a solution u(x, t)=w(xct) with the limits w( ± ∞) = u± (see Volpert [START_REF] Volpert | Elliptic Partial Differential Equations Volume 2. Reaction-diffusion equations[END_REF] and the references therein). This wave is a monotonically decreasing function of x, and it is globally asymptotically stable. The wave speed c admits various estimates and representations.

If the activation connectivity function 𝜙 a has a finite support and the inhibitory connectivity function identically equals zero, 𝜙 i (x) ≡ 0, then, as before, there exists a monotone wave solution of equation (3.1). Its existence and stability can be studied by conventional methods based on the maximum principle. [START_REF] Chen | Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations[END_REF][START_REF] Ducrot | Spectrum of some integro-differential operators and stability of travelling waves[END_REF][START_REF] Ma | Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation[END_REF] Its speed is investigated in Bessonov et al [START_REF] Bessonov | Estimate of the travelling wave speed for an integro-differential equation[END_REF] and Beuter et al. [START_REF] Beuter | Modeling Of post-stroke stimulation of cortical tissue[END_REF] Finally, if both connectivity functions are present and they have bounded supports, existence and stability of waves with the limits w( ± ∞) = u± are not studied analytically since the existing methods are not applicable. We can expect their existence and stability in the case where the homogeneous in space solution u -is stable (Section 3). Example of numerical simulations of such waves is shown in Figure 3 (left).

Time-periodic waves

If the homogeneous in space stationary solution u -becomes unstable, then a stationary periodic in space solution bifurcates from it. In the case of wave propagation, this periodic structure emerges behind the wave. Thus, we have two propagating solutions: the wave from u + to u -and the transition from u -to the periodic in space stationary solution. Depending on their speeds, they can either propagate one after another or they can merge. An example of such transition is shown in Figure 3 (right) for the response functions S a (u)=S i (u)=arctan(hu) (the values of parameters are given in the Appendix A). Here, the solution u = 0 is unstable, and we observe a transition from the unstable homogeneous in space solution to the stable periodic in space structure. This transition propagates as a time periodic travelling wave.

Stationary periodic waves

Stability analysis carried out in Section 3 shows the existence of eigenvalues with a nonzero imaginary part in the case of asymmetric connectivity functions. The corresponding eigenfunctions have the form cos(𝛽t + 𝜉x). These are waves with the spatial frequency 𝜉 and the speed -𝛽∕𝜉. They are observed in numerical simulations (Figure 4).

Other regimes

Stationary periodic waves can become unstable leading to the appearance of modulated waves. In this case, the spatial frequency remains constant but the amplitude of spatial oscillations is a periodic function itself. In the example shown in Figure 5 (left), the periodic wave propagates to the right and the modulation wave propagates to the left. For other values of parameters, the modulation of the amplitude loses its periodicity, and more complex propagating solutions are observed (Figure 5, right). Their spatial frequency remains approximately constant while the amplitude has aperiodic oscillations.

METHODS OF STIMULATION

Constant activation

In the case of monotone waves, the wave speed can change in the damaged tissue because the connectivity and response functions are different in comparison with the normal tissue. We studied the influence of constant stimulation I on the speed of wave propagation in Bessonov et al [START_REF] Bessonov | Estimate of the travelling wave speed for an integro-differential equation[END_REF] and Beuter et al. [START_REF] Beuter | Modeling Of post-stroke stimulation of cortical tissue[END_REF] It is shown that under some conditions, it is possible to restore the speed of wave propagation by the stimulation. These conditions are formulated in terms of the relative properties of the normal and damaged tissues.

Linear activation

We call the stimulation I = ku linear activation since additional activating signal is linearly proportional to the observed potential in the tissue. Hence, the same electrode measures electric potential in the cortex and sends stimulating signal depending on the measured signal. In this case, we replace the intrinsic inhibition rate 𝜎 by 𝜎k. Let us recall that in the case of periodic waves, the wave frequency 𝜉 should satisfy the condition 𝜎<Φ 2 (𝜉). This condition may not be satisfied for the damaged tissue since the connectivity and response parameters a i , b i , s a , s i can have lesser value for the damaged tissue than for the normal tissue. Denote by Φ * 2 (𝜉) the corresponding function for the damaged tissue. If the condition 𝜎<Φ * 2 (𝜉) is not satisfied but Φ * 2 (𝜉) > 0, we can choose such k that 𝜎k < Φ * 2 (𝜉). Hence, we can restore the same frequency of wave propagation. However, the wave speed c in (3.10) does not depend on 𝜎. Therefore, we cannot restore the wave speed by this method of stimulation.

Periodic activation

Due to decrease of the connectivity and response functions in the damaged tissue, periodic waves can completely disappear resulting in stability of the homogeneous in space stationary solution. Stability condition in the damaged tissue holds if 𝜎>Φ * 2 (𝜉) for all 𝜉. In this case, the periodic stimulation I = I 0 cos(px + qt) with properly chosen p and q provides periodic waves with the corresponding frequency p and speed -q∕p.

This approach can be applied in the case where the homogeneous in space stationary solution is unstable and the resulting spatial solution has a sufficiently small amplitude. Then for a strong enough stimulation function, we get the solution with the desired speed and frequency modulated by weak amplitude oscillations (Figure 6). The same stimulation applied to low amplitude modulated waves (Figure 5) gives the same result as in Figure 6 (right).

Linear deactivation and periodic activation

The periodic stimulation considered above is efficient if remaining intrinsic waves in the damaged tissue have sufficiently small amplitude and they do not perturb the stimulation. If this is not the case, we can impose, first, inhibiting stimulation I 1 =ku in order to suppress remaining oscillations. The second stimulation I 2 = I 0 cos(px + qt) will allow the reconstruction of desired wave frequency and speed. Thus, the total stimulation I = I 1 + I 2 consists of two parts indicated above.

Complete signal reconstruction

Signal propagation can be different in the normal and in the damaged tissues since their properties differ from each other. Let us write Equation 2.1 for the normal tissue [START_REF] Vosskuhl | Non-invasive brain stimulation: a paradigm shift in understanding brain oscillations[END_REF] a (u))g(u)H(W [START_REF] Vosskuhl | Non-invasive brain stimulation: a paradigm shift in understanding brain oscillations[END_REF] i (u)), (5.1)

𝜕u 𝜕t = D 𝜕 2 u 𝜕x 2 + g(K -u)H(W
and 𝜕v 𝜕t = D 𝜕 2 v 𝜕x 2 + g(K -v)H(W (2) a (v)) -g(v)H(W (2) i (v)) (5.2)
for the damaged tissue. The superscripts 1 and 2 correspond to the normal and to the damaged tissue showing that the parameters of these equations are different (connectivity and response functions). Therefore, the solutions u(x, t) and v(x, t) are also different even if the initial conditions are the same, u(x, 0)=v(x, 0) for x ∈ R. Consider, next, equation for the damaged tissue with an external stimulation I(x, t):

𝜕z 𝜕t = D 𝜕 2 z 𝜕x 2 + g(K -z)H(W (2) a (z)) -g(z)H(W (2) i (z)) + I(x, t). (5.3)
We are interested in the following question: Is it possible to choose stimulation I(x, t) in such a way that solution z(x, t) of Equation 5.3 becomes the same as solution u(x, t) of Equation 5.1?

If this is possible, then external stimulation can completely reconstruct wave propagation. In spite of the importance of this problem for the application, the solution of this seemingly difficult question is simple. Assuming that z(x, t)=u(x, t), substitute the function u(x, t) in Equation 5.3. Then we get [START_REF] Vosskuhl | Non-invasive brain stimulation: a paradigm shift in understanding brain oscillations[END_REF] a (u))g(u)H(W [START_REF] Vosskuhl | Non-invasive brain stimulation: a paradigm shift in understanding brain oscillations[END_REF] i (u))g(Ku)H(W [START_REF] Canavero | Textbook of Therapeutic Cortical Stimulation[END_REF] a (u)) + g(u)H(W [START_REF] Canavero | Textbook of Therapeutic Cortical Stimulation[END_REF] i (u)). Thus, the stimulation function [START_REF] Vosskuhl | Non-invasive brain stimulation: a paradigm shift in understanding brain oscillations[END_REF] a (u)) -H(W [START_REF] Canavero | Textbook of Therapeutic Cortical Stimulation[END_REF] a (u))

I(x, t)= 𝜕u 𝜕t -g(K -u)H(W (2) a (u)) + g(u)H(W (2) i (u)) = g(K -u)H(W
I(x, t)=g(K -u) ( H(W
) -g(u) ( H(W (1) i (u)) -H(W (2) i (u))
)

gives solution of the complete reconstruction problem. If we need to reconstruct the solution u(x, t) not for all values of x, but only in one or several points x i (electrodes), then different stimulation functions can provide such solution. In this case, we can also consider the problem of optimal control minimizing the total intensity of stimulation ∫ T 0 ∫ Ω I 2 (x, t)dxdt.

DISCUSSION

6.1 Modeling 6.1.1 Diffusion term in Equation 2.1

Neural field models are usually considered without the diffusion term. [START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease[END_REF][START_REF] Pinotsis | Anatomical connectivity and the resting state activity of large cortical networks[END_REF][START_REF] Ermentrout | Existence and uniqueness of travelling waves for a neural network[END_REF][START_REF] Chen | Wave propagation mediated by GABAB synapse and rebound excitation in an inhibitory network: a reduced model approach[END_REF] From the mathematical point of view, introduction of the diffusion term is a generalization where the particular case D = 0 reduces it to the previous model. In Bessonov et al, [START_REF] Bessonov | Estimate of the travelling wave speed for an integro-differential equation[END_REF] we showed that diffusion term influences the speed of wave propagation. In fact, there are two different regimes determined, respectively, by diffusion and by nonlocal interaction (connectivity function). If the diffusion coefficient is sufficiently large, then the wave speed is proportional to √ D, as it is usually the case for reaction-diffusion waves. If the diffusion coefficient is small enough, then the wave speed becomes independent of it, and it is determined by the connectivity function. Hence, diffusion can influence wave propagation, and it can be useful to discuss its meaning in the neural field models.

The integral term in Equation 2.1 describes signal transmission between neurons along the axons. There are other possible mechanisms which can influence neuron activation. The first one is ion diffusion through the extracellular matrix which changes the distribution of electric potential and influences neuron activity. Assuming that the latter is proportional to the ionic concentrations, we naturally describe it by the diffusion term. Similar action can be produced by gap junction communication where neurons (and possibly glia cells) exchange ions directly and not through the extracellular matrix. Finally, another mechanism is related to the ephaptic effect [START_REF] Buzsaki | The origin of extracellular fields and currents EEG, ECoG, LFP and spikes[END_REF] where neurons feel electric gradients from large groups of activated neurons through the extracellular matrix and respond to them. Diffusion term suggests a phenomenological description of this effect taking into account space transmission of neuron activation "from large to small" values.

Connectivity and response functions

Connectivity functions are usually considered in the form of exponential functions with possibly different decaying rates for the activation and inhibition. [START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinson's disease[END_REF] Nerve tissue is in general anisotropic with the length and density of axons possibly varying in space and according to their direction. The normal (orthodromic) direction of nerve impulse propagates from the soma towards the axon terminals. Therefore, the connectivity between the points x and y can differ from the connectivity between y and x. This difference signifies that the connectivity functions can be asymmetric, that is, their decay rate for positive and negative values of x can be different. Asymmetric connectivity functions are discussed in Pinotsis et al [START_REF] Pinotsis | Anatomical connectivity and the resting state activity of large cortical networks[END_REF] on the basis of the experimental data. [START_REF] Stephan | Advanced database methodology for the collation of connectivity data on the Macaque brain (CoCoMac)[END_REF] Let us also note that antidromic impulse (direction opposite to normal) can be initiated by electric stimulation. It can be considered as an additional method of post-stroke stimulation.

The introduction of two different connectivity functions (activation, inhibition) and their possible asymmetry is essentially used in this work. Periodic spatial structures and waves emerge under certain relations between the two functions. It can be formulated as short range activation and long range inhibition. Surprisingly, this condition is similar to the condition of emergence of Turing structures in the activator-inhibitor models described by the reaction-diffusion systems of equations. This similarity allows us to suggest that this can be a general rule of the emergence of spatial structures which can be realized in different models.

Let us now discuss the role of two different response functions. If S a (u) ≡ S i (u), then Equation 3.2 has typically three solutions: u + , u 0 , u -,whe r eu + and u -are stable points and u 0 is unstable. Periodic spatial structures cannot bifurcate from the stable points since they lie on the "flat" parts of the sigmoid function, and the derivatives S ′ a (u ± ) are small (cf Equation 3.7). The point u 0 cannot give rise to such structures neither because it is unstable. The situation becomes different in the case of different functions S a (u) and S i (u). Equation 3.2 can have a stable solution on growing branches of these functions (Figure A1).This point is stable if the slope of the inhibition response function is greater than the slope of the activation response function at the point of their intersection (we set here 𝜎 = 0 for simplicity).

Periodic spatial structures can be observed around solution u = 0 for a single response function S a (u)=S i (u)= arctan(hu) and 𝜙 0 i >𝜙 0 a (Figure 1 and Appendix A). The linearized problem in this case is similar to the linearized problem with two different response functions and the solution u -on the growing branch (Figure A1).

Cortex waves and patterns

A large variety of waves and patterns are observed in neural field models. Among them stationary and oscillating Turing structures, [START_REF] Venkov | Dynamic instabilities in scalar neural field equations with space-dependent delays[END_REF] wave fronts, [START_REF] Pinto | Spatially structured activity in synaptically coupled neuronal networks: I. Travelling fronts and pulses[END_REF] periodic travelling waves (TW), [START_REF] Venkov | Dynamic instabilities in scalar neural field equations with space-dependent delays[END_REF][START_REF] Atay | Neural fields with distributed transmission speeds and long-range feedback delays[END_REF][START_REF] Meijer | Travelling waves in a neural field model with refractoriness[END_REF][START_REF] Senk | Conditions for traveling waves in spiking neural networks[END_REF] modulated waves, [START_REF] Venkov | Dynamic instabilities in scalar neural field equations with space-dependent delays[END_REF] pulses, [START_REF] Venkov | Dynamic instabilities in scalar neural field equations with space-dependent delays[END_REF][START_REF] Pinto | Spatially structured activity in synaptically coupled neuronal networks: I. Travelling fronts and pulses[END_REF][START_REF] Pinto | Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses[END_REF] various two-dimensional patterns (see Kilpatrick and Bressloff [START_REF] Kilpatrick | Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression[END_REF] and the references therein). They are observed in all types of brain activities. These studies have various biological implications. Periodic waves propagation occur during language, 12,53-55 motor, [START_REF] Takahashi | Large-scale spatiotemporal spike patterning consistent with wave propagation in motor cortex[END_REF] visual, [START_REF] Sato | Traveling waves in visual cortex[END_REF] memory and spatial navigation [START_REF] Zhang | Traveling Theta Waves in the Human Hippocampus[END_REF] activities. In a review paper, Wu et al [START_REF] Wu | Propagating waves of activity in the neocortex: What they are, what they do[END_REF] examined propagating waves of activity within and between cortical areas during action, perception, or cognition. These authors suggest that TW are a manifestation of depolarization of the neuronal membrane. This depolarization increases the probability of firing action potentials in the population and these spikes will in turn depolarize more postsynaptic neurons in the neighboring area to sustain the propagation of the TW. [START_REF] Wu | Propagating waves of activity in the neocortex: What they are, what they do[END_REF] TW appear to subserve other functions as well. These authors suggest that "a sensory-evoked wave propagating to a larger area would increase the sensitivity/network gain for incoming stimulation. In this sense, the evoked wave generates an unintentional focus of attention in the sensory cortex. Furthermore, propagating waves associated with an oscillation can organize spatial phase distributions in a population of neurons." In a more recent review, [START_REF] Muller | Cortical travelling waves: mechanisms and computational principles[END_REF] it was noted that cortical TW recorded at mesoscopic or macroscopic scales can be "spontaneously generated by recurrent circuits or evoked by external stimuli and travel along brain networks at multiple scales, transiently modulating spiking and excitability as they pass."

Stimulation

We examine the situation of massive stroke at the chronic stage (6 months or more). Since the recent work of Rapela, [START_REF] Rapela | Rhythmic production of consonant-vowel syllables synchronizes traveling waves in speech-processing brain regions arXiv[END_REF][START_REF] Rapela | Traveling waves appear and disappear in unison with produced speech[END_REF] we know that when a subject produces consonant-vowel syllables, propagating waves are observed across the cortex. Rapela observed that during the production of traveling waves consonant-vowel syllables tend to propagate in the ventro-dorsal direction (ie, have negative speed). In addition, they tend to appear before the initiation of consonant-vowel syllables and disappear before their termination. In moments of silence, traveling waves tend to reverse direction and propagate in the dorso-ventral direction (ie, have positive speed) (see Figure 2; Rapela [START_REF] Rapela | Traveling waves appear and disappear in unison with produced speech[END_REF] ). Below, we consider two scenarii in which the model could be useful.

Scenario 1

In the first scenario, we consider that white matter connection is no-longer possible between two areas of convergence because of the presence of an ischemic core and a perilesional area surrounding this core zone (see Mandonnet and Duffau, [START_REF] Mandonnet | Understanding entangled cerebral networks: a prerequisite for restoring brain function with brain-computer interfaces[END_REF] (figure 2)). One solution suggested by Mandonnet and Duffau 31 is to record in vPMC (ventral premotor cortex) and stimulate in vSMG (ventral supra marginal gyrus) to create a BCBI (brain computer brain interface) between these two areas. We propose instead to place the stimulator across the partially damaged but still functional ischemic zone and to use the model presented above (and in Bessonov et al [START_REF] Bessonov | Estimate of the travelling wave speed for an integro-differential equation[END_REF] and Beuter et al [START_REF] Beuter | Modeling Of post-stroke stimulation of cortical tissue[END_REF] ) to restore the parameters value of the propagating wave. Such an intervention could facilitate long-range communication disrupted in aphasia with only one grid of electrodes. The stimulator is used to record and stimulate at the cortical level in the perilesional area to recreate de bidirectional synchronized link (Figure 7).

Scenario 2

In the second scenario, the ischemic zone is located in the vicinity of vSMG and information sent by vPMC can no longer be received in this zone. In this case, the stimulator is placed over the perilesional area. The traveling waves reach the perilesional zone and are recorded and analyzed before the same electrodes are used to stimulate the cortical tissue with parameters values modulated by the model. Again in this scenario, long range communication is facilitated (Figure 8).

How will the modulated stimulation be integrated with pathological activity? Part of the answer may come from a recent study by Voigt et al [START_REF] Voigt | Intracortical Microstimulation Modulates Cortical Induced Responses[END_REF] using Intracortical Microstimulation (ICMS). In a different context, these authors wrote that "a low-current ICMS pulse modulated the way the auditory cortex processed a peripheral stimulus, by supra-additively combining the response to the ICMS with the cortical processing of the peripheral stimulus. This shows that the response to electrical stimulation is not substituting ongoing cortical activity but is integrated into the natural processes." Of course, the context in which we propose to use model-based stimulation to treat post-stroke aphasia is different. [START_REF] Mandonnet | Understanding entangled cerebral networks: a prerequisite for restoring brain function with brain-computer interfaces[END_REF] the bidirectional link between the two areas can be restored but in this case using only one stimulator. vPMC, ventral premotor cortex; vSMG, ventral supra marginal gyrus. In this case, the alternate vSMG is progressively substituted to the original vSMG. This may be facilitated by combining neuromodulation and rehabilitation and if the treatment is successful the possibility to remove the electrode grid should be considered [Colour figure can be viewed at wileyonlinelibrary.com] The proposed cortical stimulation could assist at least temporarily the damaged tissue. We do not exclude yet the possibility that after combining modulation of cortical waves with a rehabilitation program, the electrodes might be no longer needed. Although the model is proposed in the treatment of a specific condition (ie, post-stroke aphasia), it is generic.

CONCLUSIONS AND PERSPECTIVES

Brain disorders are considered a major and growing health care problem today, notably because of the aging of the [START_REF] Dickey | Incidence and profile of inpatient stroke-induced aphasia in ontario[END_REF] Each year, more than 15 million people worldwide suffer from stroke, [START_REF] Feigin | Global and regional burden of stroke during 1990-2010: Findings from the global burden of disease study 2010[END_REF] and 1.1 million persons are affected in Europe, [START_REF] Béjot | Epidemiology of stroke in europe and trends for the 21st century[END_REF] often resulting in persisting handicaps such as language disorders [START_REF] Laska | Aphasia in acute stroke and relation to outcome[END_REF] as well as cognitive, motor, and sensory impairments. [START_REF] Gardner | Orthopedic issues after cerebrovascular accident[END_REF] Recovery can occur spontaneously but it can be amplified and accelerated by the use of noninvasive and invasive brain stimulation techniques. During the very late stage of recovery, improvements appear to reach a plateau in most patients but some patients continue to improve up to 25 years post-stroke. [START_REF] Smania | How long is the recovery of global aphasia? Twenty-five years of follow-up in a patient with left hemisphere stroke[END_REF] Thus, there is a real need to explore how brain stimulation techniques could enhance remaining cortical excitability in the affected hemisphere [START_REF] Carmichael | Brain excitability in stroke: The yin and yang of stroke progression[END_REF] and accelerate late (or very late) phase recovery as nearly half of all stroke survivors require long-term care resulting in long-term high health care costs [START_REF] Azad | Neurorestoration after stroke[END_REF] and low quality of life.

The aim of the present paper was to study a theoretical model which explores the alteration of brain waves due to post-stroke cortical damage and show that, under specific conditions, cortical electrical stimulation could compensate for these deleterious effects. We discuss different stimulation methods and the conditions of their applicability. The theoretical studies show satisfactory results but the theory should be confirmed by the clinical proof of concept with the data from patients. A number of further steps will be required for the implementation of the stimulation technique, including data treatment and the development of the brain-computer interface. Additional studies will be required in order to determine optimal and efficient patient specific stimulation based on the closed-loop approach.

From the modeling perspective, there are various directions of the development of neural field models. Among them, let us mention fractional derivative models which are used to describe biological tissues including heart and brain. [START_REF] Magin | Fractional calculus models of complex dynamics in biological tissues[END_REF] Fractional derivatives with respect to time take into account memory and delay effects, and they provide a flexible modeling approach. [START_REF] Jajarmi | A new fractional analysis on the interaction of HIV with CD4 + T-cells[END_REF] There is a vast literature devoted to nonlinear dynamics and numerical resolution of such models (see Hajipour et al [START_REF] Hajipour | On an accurate discretization of a variable-order fractional reaction-diffusion equation[END_REF][START_REF] Hajipour | On the nonlinear dynamical systems within the generalized fractional derivatives with mittag-leffler kernel[END_REF] and the references therein) which can be used for further studies of neural field problems.

. 9 )FIGURE 1

 91 FIGURE 1 Function Φ 1 (𝜉) for two different sets of parameters (left) and a stationary periodic in space solution bifurcating from the homogeneous in space solution (right). The red interval on the x-axis is used to introduce a perturbation of the homogeneous solution, the black dot at the curve is used to measure the speed of travelling waves [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2

 2 FIGURE 2 Bifurcation diagram: the amplitude of the stationary periodic in space solutions obtained in numerical simulations as a function of 𝜎 for different values of a 1 and a 3 . The curves ordered according to the values of parameters given in the inserted box from the top to the bottom. Dashed lines show the transitions between the solutions with different periodicity. Other values of parameters are given in the Appendix A

FIGURE 3 FIGURE 4

 34 FIGURE 3 Snapshots of a monotone wave (left) and of a wave with a stationary periodic structure behind it (right) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5

 5 FIGURE 5 Snapshots of periodic waves with a periodically modulated amplitude (left) and with aperiodic amplitude modulation (right). The wave of amplitude modulation moves in the direction opposite to the direction of the periodic wave [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6 A

 6 FIGURE 6 A stationary periodic in space solution (left). Applying the stimulation function I = I 0 cos(px + qt), we get a solution close to a periodic wave with a weakly oscillating amplitude. The spatial frequency and the speed of propagation are determined by the parameters p and q of the stimulation function [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7 FIGURE 8

 78 FIGURE 7 Schematic representation of scenario 1. In post-stroke aphasia we propose to restore long-range communication (interrupted by white matter ischemia) by reconnecting pre-identified areas which are still functional. Some electrodes are used to record activity coming from one area and the model is used to shape the stimulation pattern sent to the other area. Thus, as proposed by Mandonnet and Duffau 31 the bidirectional link between two areas can be restored but in this case using only one stimulator. vPMC, ventral premotor cortex; vSMG, ventral supra marginal gyrus [Colour figure can be viewed at wileyonlinelibrary.com]

ACKNOWLEDGEMENTS

The work was partially supported by the "RUDN University Program 5-100," the RSF grant number 18-11-00171, and the French-Russian project PRC2307. There are no conflicts of interest to this work.

RUDN University Program 5-100, Grant/Award Number: RSF grant number 18-11-00171; The French-Russian project, Grant/Award Number: PRC2307 considered.

APPENDIX A: FUNCTIONS AND PARAMETERS

A.1 Response functions.

Typical sigmoid functions are shown in Figure A1: S a (u)= e 2x-5 1 + e 2x-5 , S a (u)= e 5x-12. [START_REF] Takeuchi | Combinations of stroke neurorehabilitation to facilitate motor recovery: perspectives on Hebbian plasticity and homeostatic metaplasticity[END_REF] 1 + e 5x-12.5 .

FIGURE A1


Polokwane

Deutsche Fremde

Sharing knowledge through food and eating HORNSDALE DAKAR TRONDHEIM

We have changed the original title "International Evening Class" to "Transversal Evening Class" to emphasise the transversal orientation of the project. A transversal "cutting" through and across places, institutions, disciplines, to draw inter-local lines but also connections between different types of institutions-big and small, independent, academic, popular-not focussing too much on inter-nationality, a term which still carries and confirms the concept of 'nationality' despite other possible inter-connections.

Funding, Development and Continuation of TEC

As announced in the DARIAH application, submitted in October 2020, we were aiming for a large scale, multi-local project that would happen at different places simultaneously and independently; carried out by different partners (such as the artist collective On-Trade-Off/Lumbumbashi-Brussels, Z33/Hasselt, Framer-Framed/Amsterdam, the filmmaker Jean-Pierre Bekolo at Quartier Mozart/Yaoundé, Assumpta Mugiraneza at Iriba Center/Kigali), over a period of, at least, two years. This comprehensive application (in collaboration with Harun Farocki Institute/Berlin), designed for the German TURN fund / Bundeskulturstiftung (including 22 letters of intent from collaborating artists and institutional partners, including an invitation by CAMP-notes on education for documenta fifteen), was not selected for funding, although we received very positive feedback.

Nevertheless, conceptualising, planning and describing the project in detail, including intense communication with the respective partners (Nov 2020-March 2021) was an important step for the actual project development and for successive successful local and smaller fundraising-applications, such as Trondheim Kommune, Goethe Institute Kigali, International Film Festival Oberhausen, and ifa-institute for foreign affairs.

Scaling (space) / Stretching (time)

Even though, the envisioned large-scale project taking place at several places in parallel, condensed to "synch" moments exhibiting intermediate results, would have created more visibility, the now more processoriented structure certainly has its advantages. Running projects in consecutive allows for testing, learning form previous classes, adapting themes, and for a for step-by-step sustainable approach. The luxury of time cannot be overstated. Also to involve former participants and institutional partners for future evening classes, on the basis of experienced collaborations.

More recently, the planned evening class events in Hanau/Germany (spring 2023) has been approved for funding. We are also about to write the next application for the Globus Call of the Nordisk Kulturfond (deadline Oct 2022), for a project idea which developed out of a "mid-air" class that crystallises the proposed

Evening Class as (Artistic) Methodology

"En rachâchant" is an invented expression. According to the critic Yann Lardeau in Cahiers du Cinéma at the time, "it onomatopoeically hints at harping on, harking back, buying back, muttering, mumbling, chewing, knowing, fretting, fuming and murdering!" "By re-de-de re-see-see re-pee-pee-ting!" was a translation suggested by Ina C. Jaeger and Ciba Vaughan.

Classes on site: Trondheim, Kigali, Oberhausen

Kigali: Dutaramane-Buhanga | Invitation to evenings dedicated to sharing knowledge

The documentation / outcome of the Evening Class in Kigali, that evolved around the notion of the Untranslatables is not yet published online, it needs further careful sound-editingSince we (Stacy Hardy, Marie-Hélène Gutberlet and I) left, after one week of intense discussions, presentations, encounters, a comprehensive research has been developing in relation to sorghum (millet) and its agri-cultural practice and history in Rwanda, its social implications and layered connotations and meanings in Kinyarwanda. Towards a "Sorghunnaire," a multi-media dictionary wrapped around disappearing knowledge, not only connected to food and beer production but also to language. The "Sorghunnaire," will find its form on the woa.kein.org website as well, carried out collectively with the same group of people who gathered at IRIBA Centre in November, invited by Assumpta Mugiraneza (main funding comes from Goethe Institute Kigali).

Of all evening classes so far, the methodology as described above had the most surprising and poignant effect, or maybe impact on the TEC participants in Kigali. The IRIBIA centre for Multimedia Heritage, founded and headed by Assumpta Mugiraneza, has generated an impressive, cross-generational community of dedicated researchers with diverse working backgrounds, from law to theatre, from students of computer science to political and social science. What holds them together is, to a large extent, the driving force, Assumpta, as well as the search to passionately go beyond their own professions. If open mindedness is considered a skill, then this could be described as their ongoing training programme.

There is always a lingering trap, coming from outside and especially from Europe, to create/set-up a workshop-like situation in a place unknown to the visitors. The contested patriarchal attitudes and relations have a long history and so many facets that it's not an easy task to avoid or undo these. However, there is an urgent need to stay in conversation and to create frameworks for these decolonising conversations. The collaboration with Stacy Hardy, a writer and editor from South Africa already opened up the binary northsouth constellation, but most surprisingly for the participants, was that we didn't come with a "topic" or a "curriculum", but only with a methodology. "You are the curriculum," we hear Stacy's voice over in the video. The notion of the "untranslatable" in a place like Rwanda where language and speech is the core of all art, was an invitation to bring an object or a word that seemed impossible to translate. And indeed, each word written on the blackboard could open up long and vivid conversations, leading to stories and misunderstandings. It brought much laughter, but at the same time, the trauma of the Genocide remained a profound element of the difficulty or impossibility to translate. While these conversations were again constantly translated within the group, switching between French, English, Kinyarwanda and its neighbouring Burundian variations, sometimes even German or Afrikaans came into play.

After some time of confusion about the methodological openness of what could actually be learned from this gathering, it set a creative and liberating process of appropriation in motion, to spend time with questions, things, words, meanings that actually matter, instead of learning things not known. And for us, it became an invaluable gift to participate in this process and learn so many things we didn't know we even wanted to know.

What happens if we take seriously what we do know ?

Trondheim: Slow Education

How do we learn, engage and communicate through and with music? How does it shape our views, attitudes and ways of thinking? The evening class Slow Education (see also video) consisted of a very different format, and offered a series of public listening sessions led by researchers, musicians, enthusiasts, and artists who narrated with and through music. The Trondheim folkebibliotek's truly international vinyl selection served as a starting point to invite, present and share an array of listening habits and tastes, to unravel polyphonous narratives of and through popular and unpopular culture. (Together with Prerna Bishnoi).

Lisa Størseth Pettersen: Is it the Sea? One song on loop. Carried to a place where it could belong to.

Tested in various ways, stretched over a few hours, until it gets dark and a choir takes over. going on a revolutionary sounding, historic world tour, but also making, creating worlds, with a series of Intros leading to an Outro. This evening class listening session followed some personal, some environmental and

Trondheim: Toward a Third

The Evening Class Toward a Third, had a different set-up, context and purpose. It was a series of screenings, readings and debates which happened at KIT/NTNU, partly with students of the Art Academy, partly with participants / visitors across faculties, mainly from the media and film department. This evening class really opened up conversations, connections and lasting relations, something that rarely happens in the hecticness of "everyday businesses".

The class also had screenings of films that are usually not part of the curriculum and could build a different common ground of references among the students. Toward a Third also served as preparation and a sort of "testing/preview" for the theme-programming during the International Film Festival in Oberhausen (Synchronize! Pan-African Film Networks), but also as an experimental format, to combine readings with screenings, to offer different relations between theory, text, image, rhythm, information, contemplationinstead of talking about a film, it was rather a kind of talking with, like introducing an additional layer. Not least, having the filmmaker Ousmane Sembène in mind who loved to speak about cinema as the best evening class to educate people.

Oberhausen

Women on Aeroplanes was invited to curate this year's theme programme at the renowned International Short Film Festival in Oberhausen (May 2022). In the large scale version of the TEC concept, this venue and occasion was designed as an In Synch moment, for the various participating artists spread across different places to gather in Oberhausen. We could still realise this idea and turn the programme we were invited to curate (8 programme slots à 2 hours, plus 2 podium discussions and one youth-programme) into a large gathering and evening class as well. The programme itself had a focus on networks, working relations and conditions, and highlighted the importance of festivals in relation to African filmmaking. In this sense, we followed again the above described methodology of the evening class, to research what we are doing and to realise what we are researching.

The evening class framework allowed to "co-own" the commissioned programme of the film festival, to use the festival itself again as a site of production, and not only of representation. We could for example realise a long group-interview with young female filmmakers coming from six different countries, Salimata Bâ (Nouakchott), Yousra El-Gazzar (Cairo), Lina Alabed (Beirut), Rim Harrabi (Tunis), Amelia Umuhire (Berlin) and others. With over 15 guests the theme programme almost became a festival in itself, and it really created the vivid networking dynamic we aimed and hoped for. A gathering that could build the ground for new projects.

Through the web-platform we could also provide an extended, long lasting web-presence, design the projects' own visibility instead of disappearing in the overall branding of the film festival.

See detailed programme including guests: https://woa.kein.org/Oberhausen

The evening class "Toward a Third" transformed on the same site as Mid-Air" as a column on the right, built in

  m i d a i r B : s h e e p m e t h o d o lo gi es W h a t d o w e s e e a n d h o w d o w e d e s c r i b e i t ? a i r A : s p a t i a l i m a g i n a t i o n s / V o r s t e l l u n g s v e r m ö g th e f u t u r e a n d i t w o r k s - n d e r s t a n d p r o c e s s e s o f t r a n s i t i o n t h r o u g h t h e m a l y s i n g m a l e p o w er politi cs c ://woa.kein.org/node/3 Classes on site: https://woa.kein.org/eveningclass Video: https://woa.kein.org/node/38 Slow Education: https://woa.kein.org/Trondheim Synchro System: https://woa.kein.org/Oberhausen

The

  Transversal Evening Classes were and are neither enclosed in classrooms, nor do they take place online. They are actually anything but a class as we know it but raise all kinds of class-related questions, aiming to understand the world and our neighbourhoods from the thick of things and not from a survey perspective. An Evening Class, with all its legacy, opens up an entry point to engage in conversations and a discourse on the new challenges of divisions of labour, re-skilling and open access to education. It takes the tensions between the global and local seriously, creates new methodologies rooted in chance encounter, random perusal, reading as pleasure, "following sheep around" , clumsiness, failure and untranslatability as gifts, developing syllabi through radical rereadings of history and new imaginings of the future that go beyond our imagination. The Transversal Evening Classes realised on site, with the support of DARIAH funding and within the extended time frame (January 2021-Juni 2022), in Trondheim, Kigali and Oberhausen were: Slow Education (Sept 2021-Jan 2022, together with Prerna Bishnoi, in collaboration with the Trondheim public library), Toward a Third -Traces of a Manifesto (Oct 2021 -Jan 2022, together with Joen Vedel, in collaboration with KIT/NTNU), Dutaramane-Buhanga (Kigali, Nov-Dec 2021 with Assumpta Mugiraneza/IRIBA Center, Stacy Hardy/ Chimurenga & PASS, SA and Marie-Hélène Gutberlet/WoA), Synchro System (May 2022, together with Marie-Hélène Gutberlet in collaboration with International Film Festival Oberhausen).

(

  https://woa.kein.org/node/16) Chicks on Speed & Unnur Andrea: Cracking the Playlist. To crack the algorithms you just have to make a choice. Making a choice can make you listen what you might have never heard before. Look into the glitch, the open space, carry on the message that may have started 60 years ago into the future. Support your local library, fill your frontal lobe and the rest of your brain with pleasurable and disturbing sounds of cultural agitation. Crack the playlist now. Don't let your eardrums collect dust like the unread books in the library. (https://woa.kein.org/node/33) Ahmed (INTERPRT) & Annett Busch (Women on Aeroplanes) (7 Dec) **Bonus TRACK at Mr. FIX St. Olav's Gate 1 (TBD) For more: http://woa.kein.org/ RIGHT AROUND THE WORLD Martinus Suijkerbuijk: Perception in 3.0. Acts -Complexity, Entropy, Information. In this evening class on the structures of sound and music, the records of Pharmakon, Aphex Twin and Kraftwerk operated as information systems. Their (musical) analysis turned into an investigatory means of phenomenal perception and the structure of reality in order to proclaim music's capacity for philosophical scrutiny. Scientific icons like noise, entropy and information were dissected and positioned as performing cultural roles such as, bridging art and science with the micro-and macro expressions of music. (https://woa.kein.org/node/ 35) Oscar Debs: Electronic Contrasts. The session tackled different facets of the renowned composer Arne Nordheim's musical language.'Solitaire' is one of his most known electronic pieces, 'Lux et Tenebrae' is a concert version from Nordheim's ambitious contribution to the Scandinavian pavilion at EXPO'70 in Osaka. (https://woa.kein.org/node/34) Martin Palmer & Kyrre Laastad: Cover Music. Over three pieces by The Velvet Underground, Robert Ashley and Jenny Hval, and through a selection of record covers, this listening session unfolded the background-foreground relations that construct the complexity of an album: cover aesthetics, lyrics, volume, and modes of narration emerging between sounds, meaning and overtones.(https://woa.kein.org/node/36) Nabil Ahmed & Annett Busch: .. because that's not something you learn at school. Faire du monde,

  relation to the on-site programme in Oberhausen. What is still missing are the edited audio files of conversations, For me, the cinema is the best evening school. It not only enables me to do more and to go further than literature, but it also lets people speak in their own language-Wolof in this case.[...] We are doing something here that goes beyond our imagination. (Ousmane Sembène, 1973) O'Laughlin's daughter Ruth Castel-Branco, a labour organiser, now a research manager of the Future of Work(ers) research project at the Southern Centre for Inequality Studies, University of the Witwatersrand in Johannesburg, connects to that legacy, but being of a new generation, with a different perspective and agency. We are also in touch with Remi Hess, the biographer of HL who has been working with him since the 1970's until his death and was one the main driving forces behind republishing some of the most important HL books. And with Stuart Elden, Professor at Warwick who published, edited, translated and introduced important texts to English speaking readers. The artistic academic Elisa Bertuzzo, the theory-artist Kodwo Eshun and the creative research-writer Stacy Hardy are three other close collaborators in this endeavour.To present/publish the results and conceptual ideas of the work/research that has been done so far, and to finish the mid-air interface on the website which will show intermediate results of the open-ended process in a kind of unfolding montage, that can be continuously expanded, we still need a few more weeks.