Well-Posedness and Regularity of Solutions to Neural Field Problems with Dendritic Processing

Abstract

We study solutions to a recently proposed neural field model in which dendrites are modelled as a continuum of vertical fibres stemming from a somatic layer. Since voltage propagates along the dendritic direction via a cable equation with nonlocal sources, the model features an anisotropic diffusion operator, as well as an integral term for synaptic coupling. The corresponding Cauchy problem is thus markedly different from classical neural field equations. We prove that the weak formulation of the problem admits a unique solution, with embedding estimates similar to the ones of nonlinear local reaction–diffusion equations. Our analysis relies on perturbing weak solutions to the diffusion-less problem, that is, a standard neural field, for which weak problems have not been studied to date. We find rigorous asymptotic estimates for the problem with and without diffusion, and prove that the solutions of the two models stay close, in a suitable norm, on finite time intervals. We provide numerical evidence of our perturbative results.

1 Introduction

The dynamics of brain activity is intrinsically nonlocal: at microscopic level, the cellular nucleus of a neuron is in its soma, whose average diameter is 20 micrometres; its dendritic branches, collecting input from other neurons, have an extent variable between 100 micrometres and 1 mm ((Harris and Spacek 2016), Table 1.1); its axons, which transmit activity to other neurons, have terminations as far as 1 m away from the soma (Ma and Gibson 2013).

Coarse-grained descriptions of large-scale neuronal activity inherit this nonlocality, and model long-distance cortical connections via integral terms. The simplest and most popular of these models is the neural field equation

$$\begin{aligned} \begin{aligned}&\partial _{t}v(x,t) = -\gamma v(x,t) + \int _{\Omega } W(x,x^{\prime })S(v(x^{\prime }))dx^{\prime } + G(x,t),{} & {} (x,t) \in \Omega \times [0,T],\\ {}&v(x,0) = v_0(x),{} & {} x \in \Omega . \end{aligned} \end{aligned}$$

(1.1)

In this model, v(x, t) represents the macroscopic voltage at time t and position x in the continuum cortex $\Omega$. The voltage decays at rate $\gamma$, and is influenced by the nonlocal integral coupling, and by the external input G. The former collects contributions from the whole cortex: the firing rate function S acts as a nonlinear gain, and the synaptic kernel W models connections from point $x'$ to point x in the cortex.

Since their introduction by Wilson and Cowan (1973), and by Amari (1977), neural fields have been used to model large-scale patterns of activity (see the textbooks and monographs (Ermentrout 1998; Ermentrout and Terman 2010; Bressloff 2012, 2014; Coombes et al. 2014; Coombes and Wedgwood 2023)). Mathematical neuroscientists have progressively developed dynamical systems methods to study the wide variety of nonlinear patterns supported by neural field equations, including localised bumps, travelling waves, and rotating waves (Ermentrout and McLeod 1993; Folias and Bressloff 2005; Gils et al. 2013; Kilpatrick and Ermentrout 2013; Meijer and Coombes 2014; Laing 2014; Kilpatrick and Faye 2014; Visser et al. 2017; Schmidt and Avitabile 2020). The analytical treatment of these evolution equations was initiated in the 1990s, by proving the existence of travelling wave solutions using perturbative arguments (Ermentrout and McLeod 1993). In addition, the well-posedness of (1.1) has been studied with functional analytical methods, as a Cauchy problem on the spaces of continuous functions (Potthast and beim 2010) and square integrable functions (Faugeras et al. 2008) defined on the cortical space $\Omega$.

Since solutions in closed form are available only in simple cases, numerical methods have been proposed for one- and two-dimensional cortices. These algorithms must overcome the problem of evaluating efficiently the nonlocal term, and schemes that leverage rank reduction (Lima and Buckwar 2015), fast Fourier transforms over $\Omega$ (Rankin et al. 2013), or over [0, T] (Hutt and Rougier 2010) have been proposed. Recently, collocation and Galerkin, finite elements, and spectral schemes have been analysed systematically, as abstract projection schemes for neural fields, seen as Cauchy problems on Banach spaces (Avitabile 2023).

The neural field Eq. (1.1) is now a well-established phenomenological model for neurodynamics in the cortex, and has been extended in several ways. Models with multiple populations (Pinto and Ermentrout 2001) and with transmission delays (Faye and Faugeras 2010; Meijer and Coombes 2014) have been proposed in the 2000s, whereas systems incorporating time-dependent synaptic kernels to model short-term plasticity are becoming available at the time of writing (Cihak and Kilpatrick 2023). As new models are conceived, novel mathematical approaches are developed to analyse them: neural fields with delays, for instance, led to the development of a dedicated theory based on sun–star calculus (Visser 2013; Gils et al. 2013; Visser et al. 2017).

Models with nonlocal terms arise also in other branches of the life sciences. In particular, a wide range of nonlocal reaction–diffusion equations have been studied to describe specific aspects of the evolution of biological populations (see (Volpert 2014), Sect. 1.2.4, Chapter 4 and references therein). In such models, integral terms arise naturally when modelling populations compete for global resources, speciation, and cooperation in reproduction. In this context, a notable example is the nonlocal Fisher-KPP equation (Berestycki et al. 2009; Perthame and Génieys 2007). Nonlocal models are also found in models of tumour growth, as a consequence of variations in nutrients or oxygen concentration, or to model chemotherapy and immune system responses (Carrère and Nadin 2020). Recently, integro-differential equations have been proposed to model the emergence of cities and urban patterning (Whiteley et al. 2022). The growing literature on the subject has shown that a rigorous theoretical analysis is paramount to capture the nonlocal behaviour of solutions with respect different biological parameters. Important investigations of nonlocal reaction–diffusion theory can be found, for instance, in Bouin et al. (2020), Perthame and Génieys (2007).

The present article concerns the theoretical treatment of one of the more recent extensions to the neural field equations, which include a diffusive term. Neural fields with diffusive terms have been analysed in models with delays (Spek et al. 2020): in this context, the powerful machinery of sun–star calculus has been exploited to tackle the diffusive term. Recently, two models without delays have been proposed: in the first one, an anisotropic diffusion term emerges in the voltage Eq. (1.1), upon modelling dendrites in the tissue (Avitabile et al. 2020); in the second one, a neural field for voltage dynamics is coupled to a reaction–diffusion equation for potassium dynamics, to model spreading depression (Baspinar et al. 2023).

The latter models have an interesting and unexplored mathematical structure. The diffusion-less neural field Eq. (1.1) admits classical solutions, and the technical apparatus to analyse them relies heavily on the compactness, boundedness, and Lipschitz properties of the integral operator. In models with diffusion, the scenario changes considerably, owing to the presence of unbounded differential operators. The motivation behind the present article is that, in models with diffusion and without delays, one should be able to study solutions using classical methods for PDE analysis, and view the integral term as a nonlinear, well-behaved perturbation of a parabolic problem.

Vanishing diffusion problems are of great importance also outside of mathematical neuroscience, in the mathematical and physical communities. For instance, such problems model atmospheric/oceanographic flows and landscape evolution when fluvial erosion dominates over smoothing tendencies of the soil diffusion; further, artificial vanishing diffusion is used for stabilisation of numerical simulations. An outstanding problem in fluid mechanics is the control of boundary layer turbulence: small forces of viscous friction may perceptibly affect the motion of fluids, even in fluids with small viscosity, such as water and air. Knowledge of the behaviour of solutions for small viscosities is crucial for understanding turbulence phenomena, which have strong consequences in branches of engineering dealing with car and aircraft productions, turbine blades, and nano-technology. We refer to (Chemetov and Cipriano 2013a, b, 2014a, b, 2018) for mathematical studies on problems with vanishing viscosity.

Fig. 1

a Sketch of the cortical domain $\Omega = \cup _{\xi \in [0,L]} \Sigma _\xi$, in which $\Sigma _{L/2}$ is the somatic layer. Dendrites are modelled as a continuum of vertical, unbranched fibres, with synaptic connections to other dendrites. b In Avitabile et al. (2020), and in the numerical experiments of the present paper, we take $\Sigma _{L/2} \subset \mathbb {R}$, with kernel specified by (1.4). This kernel prescribes localised connections from a small neighbourhood of the somatic layer $\xi = L/2$, to a neighbourhood of the somatic coordinate $\xi = \xi _c$, with strength dependent on the mutual distance between points, projected on the x-axis. This modelling assumption is used in simulations, but not for the theoretical results, which work for generic kernels, and with somatic layers of arbitrary dimensions

1.1 Model with Anisotropic Diffusion, and Main Results of the Paper

We demonstrate this strategy on the model proposed in Avitabile et al. (2020), in which the cortex has an embedded anisotropy, owing to its being split into somatic and dendritic directions. With reference to the sketch in Fig. 1, the cortex $\Omega \subset \mathbb {R}^3$ is composed of dendritic fibres, aligned vertically to $U =(0,L) \subset \mathbb {R}$, with coordinate $\xi$. Dendritic fibres are unbranched, and afferent to somas, which form a continuum somatic layer, a 2-manifold $\Sigma _{L/2} \subset \mathbb {R}^3$. The subscript in $\Sigma$ indicates that the somas belong to a medial layer located at position $\xi = L/2$, and it hints that the cortex is conceived as a foliation $\Omega = \cup _{\xi \in U} \Sigma _\xi$ of which the somatic layer is the leaf by $\xi = L/2$.

In Avitabile et al. (2020), the somatic layer was assumed to be the 1-torus $\Sigma = \mathbb {T}$, whereas in the present paper we treat the more general case $\Sigma = \mathbb {T}^n$, for some $n \in \{ 1,2 \}$. The cortex is thus specified as a tensor product, with coordinates $(x,\xi ) \in \Omega = \mathbb {T}^n \times U$. Finally, for notational convenience, we introduce the sets $U_T = U \times [0,T]$, and $\Omega _T=\Omega \times [0,T]$.

The somato-dendritic model differs from a standard neural field model, in that the voltage $v(x,\xi ,t)$ propagates according to a cable equation along the dendritic fibre direction $\xi$ (see, for instance, (Tuckwell 2006)), with an external input G and a nonlocal input current F from the somatic layer,

$$\begin{aligned} \begin{aligned}&\begin{aligned} \partial _t v(x,\xi ,t) = (-\gamma + \nu \partial _{\xi }^{2})v(x,\xi ,t)&+ G(x,\xi ,t) \\&+F(v({\hspace{1.111pt}\cdot \hspace{1.111pt}},{\hspace{1.111pt}\cdot \hspace{1.111pt}},t))(x,\xi ), \end{aligned}{} & {} (x,\xi ,t) \in \Omega _T, \\&\partial _\xi v(x,0,t) = 0, \qquad \partial _\xi v(x,L,t) = 0,{} & {} (x,t) \in \mathbb {T}^n \times [0,T],\\&v(x,\xi ,0) = v_0(x,\xi ){} & {} (x,\xi ) \in \Omega , \end{aligned} \end{aligned}$$

(1.2)

where periodic boundary conditions in x are tacitly implied by taking $x \in \mathbb {T}^n$, and where the nonlocal term is given by

$$\begin{aligned} F(u)(x,\xi ) = \int _{\Omega } W(x,\xi ,x',\xi ') S(u(x',\xi '))\, dx' d\xi '. \end{aligned}$$

(1.3)

In passing, we note that Neumann boundary conditions have been chosen to model no-flux across the leaves at $\xi = 0,L$, but other choices are also possible.

Somatic currents are modelled through the nonlocal operator F. In Avitabile et al. (2020), the firing rate S is taken to be a bounded monotone smooth function, in line with neural field literature. In addition, the synaptic kernel

$$\begin{aligned} W(x,\xi ,x',\xi ') = w(\Vert x-x'\Vert _2)\delta _\rho (\xi - \xi _\text {c})\delta _\rho (\xi '-L/2), \end{aligned}$$

(1.4)

where $w :\mathbb {T}\rightarrow \mathbb {R}$, and $\delta _\rho :\mathbb {R}\rightarrow \mathbb {R}$ is a function supported on $(-\rho ,\rho )$ for some $\rho >0$, is used to model currents generated in a neighbourhood of the somatic layer, at $\xi = L/2$, and transferred to contact points in the neighbourhood of the leaf at $\xi = \xi _\text {c}$ (see also Fig. 1b). Concurrently, activity is allowed to propagate within a leaf, via the distance-dependent function w.

Model (1.2) is anisotropic: instead of the Laplacian operator $\Delta =\partial ^2_{x} + \partial _{\xi }^{2}$, the model features diffusion only along the fibre coordinate $\xi$. In addition to this local anisotropy, which is not removable, the synaptic kernel may induce a nonlocal anisotropy, as is the case for (1.4). The paper (Avitabile et al. 2020) proposes and analyses a bespoke numerical scheme for (1.4), based on differentiation matrices for the differential operator, and a fast quadrature scheme for the integral operator. This scheme exploits both anisotropies to evaluate efficiently the right-hand side and time step the system with an implicit–explicit method. Assuming well-posedness of the problem, existence, and regularity of its solutions, it was shown that (1.2) supports anisotropic travelling waves, with propagating speed along the leaves, as well as Turing-like instabilities.

The present paper provides a functional analytic treatment of (1.2), for generic choices of S, G, $v_0$, and W, that is, without assuming the anisotropic kernel (1.4). The main results of the paper can be summarised as follows:

• 1. Under mild, biologically relevant assumptions on W and S, the operator F is Lipschitz on an appropriately defined function space, and it perturbs boundedly the parabolic, anisotropic problem $\partial _t v = (-\gamma + \nu \partial _\xi ^2)v$ with mixed Neumann/periodic boundary conditions.

• 2. The weak form of (1.2) admits a unique solution, with embedding estimates similar to the ones of a nonlinear reaction–diffusion problem with local nonlinear forcing.

• 3. The analysis of weak solutions $v_\nu$ to (1.2) for $\nu >0$ relies on studying weak solutions v to the singular limit $\nu = 0$ of the problem, that is, a standard neural field posed on $\Omega$, which does not require specifications of boundary conditions. In so doing, we derive also sharper estimates on standard neural field problems, with respect to the ones currently available in the literature.

• 4. We prove that solutions $v_\nu$ depend continuously on the parameter $\nu$ and, demanding additional regularity of the problem data, that the difference $v_\nu - v$ is an $O(\nu ^{1/2})$ as $\nu \rightarrow 0$ in an appropriate norm.

• 5. We produce numerical evidence of the results above using the kernel and numerical scheme studied in Avitabile et al. (2020).

The analysis presented here provides the necessary theoretical foundation for developing numerical schemes on realistic geometries, employing, for instance, finite-elements discretisation, or multiscale schemes for weakly diffusive problems. Importantly, it seems possible to adapt the analysis presented below, with minor modifications, to the case of generic $\Omega$ with curved leaves $\Sigma _\xi$, or to neural fields with multiple populations and isotropic diffusion, such as the ones in Baspinar et al. (2023).

The paper is structured as follows: in Sect. 2 we set up the abstract formulation of system (1.2); we study the singular problem ($\nu =0$) in Sect. 3, and the regular problem ($\nu >0$) in Sect. 4; we prove continuous dependence of the solutions on $\nu$ in Sect. 5, and the $O(\nu ^{1/2})$ scaling in Sect. 6; we present numerical experiments in Sect. 7, and we conclude in Sect. 8.

2 Preliminary Assumptions and Results

We begin by describing the appropriate function spaces that will be used in the analysis of our problem.

For a given topological space ${\mathbb {X}}$, we denote by $L^{2}({\mathbb {X}})$ the space of measurable square integrable functions $u:{\mathbb {X}} \rightarrow {\mathbb {R}}$ with the usual inner product $({\hspace{1.111pt}\cdot \hspace{1.111pt}},{\hspace{1.111pt}\cdot \hspace{1.111pt}})$ and the norm $\Vert u\Vert _{L^{2}({\mathbb {X}})}=\sqrt{(u,u)}$.

Let X be a real Banach space with norm $\left\| {\hspace{1.111pt}\cdot \hspace{1.111pt}}\right\| _{X}.$ We shall denote by $X^*$ the dual of X, and by $L^{q}(0,T;X)$, $q \in [0,\infty ]$, the space of X-valued measurable functions endowed with the norm

$$\begin{aligned} \begin{aligned}&\Vert u\Vert _{L^{q}(0,T;X)} = \biggl ( \int _{0}^{T}\Vert u(t)\Vert _{X}^{q}\biggr ) ^{1/q}<\infty ,{} & {} 1\le q<\infty , \\&\Vert u\Vert _{L^{\infty }(0,T;X)} =\sup _{t\in [0,T]}\Vert u(t)\Vert _{X}<\infty ,{} & {} q=\infty . \end{aligned} \end{aligned}$$

We make the following standing assumptions:

Hypothesis 2.1

[General assumptions]

• 1. The cortical domain is $\Omega = \mathbb {T}^n \times U$, where $U \in (0,L)$, $L>0$.

• 2. The synaptic kernel W is a function in $L^2(\Omega \times \Omega )$.

• 3. The firing rate $S :\mathbb {R}\rightarrow \mathbb {R}$ is a bounded and everywhere differentiable Lipschitz function.

Further regularity assumptions on the forcing G and the initial condition $v_0$ will be provided in the statements of the upcoming results. In preparation for the abstract set-up of the problem, we derive useful properties of the integral operator

$$\begin{aligned} F(u)(x,\xi ) = \int _{\mathbb {T}^n \times U} W(x,\xi ,x',\xi ') S(u(x',\xi '))\, dx' d\xi '. \end{aligned}$$

(2.1)

Lemma 2.2

[Boundedness an Lipschitzianity of F] Under Hypothesis 2.1, the operator F defined in (2.1) is on $L^2(\Omega )$ to itself, and there exists a constant $K_F >0$ such that, for any $u, v \in L^2(\Omega )$

$$\begin{aligned} \Vert F(u) \Vert _{L^2(\Omega )} \le K_F, \qquad \Vert F(u) - F(v) \Vert _{L^2(\Omega )} \le K_F \Vert u - v \Vert _{L^2(\Omega )}. \end{aligned}$$

Further for any $q \in [1,+\infty ]$, any $u,v \in L^q(0,T;L^2(\Omega ))$, and any $t \in [0,T]$

$$\begin{aligned} \Vert F(u(t)) \Vert _{L^2(\Omega )} \le K_F, \qquad \Vert F(u(t)) - F(v(t)) \Vert _{L^2(\Omega )} \le K_F \Vert u(t) - v(t) \Vert _{L^2(\Omega )}. \end{aligned}$$

Proof

The results follow standard arguments, summarised in Avitabile (2023), and references therein, in particular (Faugeras et al. 2008). Denote by N the Nemytskii operator $N(u)(x) = S(u(x))$, and by A the linear integral operator with kernel W, so as to write $F(u) = (A \circ N)(u)$.

By (Avitabile 2023, Lemma 2.5), the operator $N :L^2(\Omega ) \rightarrow L^2(\Omega )$ satisfies the homogeneous bound $\Vert N(u) \Vert _{L^2(\Omega )} \le |\Omega |^{1/2} \Vert S \Vert _{\infty }$, and is Lipschitz with constant $\Vert S' \Vert _{\infty }$. Further by ( Avitabile (2023), Lemma 2.6), $A :L^2(\Omega ) \rightarrow L^2(\Omega )$ is bounded with $\Vert A \Vert \le \Vert w \Vert _{L^2(\Omega \times \Omega )}$.

Putting this together, it holds for all $u \in L^2(\Omega )$

$$\begin{aligned} \Vert F(u) \Vert _{L^2(\Omega )} \le \Vert A \Vert \Vert N(u) \Vert _{L^2(\Omega )} \le \Vert w \Vert _{L^2(\Omega \times \Omega )} |\Omega |^{1/2} \Vert S \Vert _{\infty }, \end{aligned}$$

and for all $u,v \in L^2(\Omega )$

$$\begin{aligned} \begin{aligned} \Vert F(u)-F(v) \Vert _{L^2(\Omega )} \!\le \! \Vert A \Vert \Vert S(u)-S(v) \Vert _{L^2(\Omega )} \!\le \! \Vert w \Vert _{L^2(\Omega \times \Omega )} \Vert S' \Vert _{\infty } \Vert u-v \Vert _{L^2(\Omega )}. \end{aligned} \end{aligned}$$

The statement is proved by setting

$$\begin{aligned} K_F = \Vert w \Vert _{L^2(\Omega \times \Omega )} \max \bigl ( |\Omega |^{1/2} \Vert S \Vert _{\infty }, \Vert S' \Vert _{\infty } \bigr ), \end{aligned}$$

and noting that any function in $L^q(0,T;L^2(\Omega ))$ takes values in $L^2(\Omega )$. $\square$

2.1 Abstract Problem Set-Up

Equipped with the operator F of the previous sections, we now consider two abstract problems. The first one is a rewriting of (1.2), the neural field problem with anisotropic diffusion posed on $\Omega = \mathbb {T}^n \times U$,

$$\begin{aligned} \begin{aligned}&\partial _t v = \nu \partial _\xi ^2 v - \gamma v + F(v) + G{} & {} \text {on }\mathbb {T}^n \times U \times (0,T], \\&\partial _\xi v = 0{} & {} \text {on }\mathbb {T}^n \times \partial U \times (0,T], \\&v = v_0{} & {} \text {on }\mathbb {T}^n \times U \times \{t = 0\}. \end{aligned} \end{aligned}$$

(2.2)

The second one is a Cauchy problem obtained from (2.2) by setting $\nu = 0$ and removing Neumann boundary conditions in $\xi$

$$\begin{aligned} \begin{aligned}&\partial _t v = -\gamma v + F(v) + G{} & {} \text {on }\mathbb {T}^n \times U \times (0,T], \\&v = v_0{} & {} \text {on }\mathbb {T}^n \times U \times \{t=0\}. \end{aligned} \end{aligned}$$

(2.3)

The problem is posed on a tensor product domain between the cortex $\mathbb {T}^n$ and the dendritic segment U, but it does not account for dendrites as cables (and there is no diffusion in the problem). This model is a classical neural field. Existence of classical solutions to this problem has been studied by several authors from a functional analytic viewpoint, Faugeras et al. (2009), Potthast and beim (2010), Avitabile (2023). In the present article, we will discuss weak solutions to the problem, as this provides the appropriate functional analytical set-up to compare solutions to the problem with anistropic (dendritic) diffusion.

We will colloquially refer to (2.2) as the regular problem $(\nu > 0)$, and to (2.3) as the singular problem ($\nu = 0$). We are interested in characterising solutions to the former as $\nu$-perturbations of solutions to the latter.

3 Existence of Solution to the Singular Problem, $\nu =0$

Our starting point is a result descending from the classical theory described in Evans (2022), Mikhailov (1978), and concerning the solution of a linear ordinary differential equation in the Sobolev space $H^{1}(0,T;\mathbb {R})$. The results presented in this section offer an accessible entry point to the ones of the following sections, with respect to which they have a similar approach, but fewer technical details.

Lemma 3.1

For any $\gamma \in \mathbb {R}_{\ge 0}$, $g\in {\mathbb {R}}$, and $f\in L^{2}(0,T;\mathbb {R})$, there exists a unique solution $v\in H^{1}(0,T;\mathbb {R})$ to the linear inhomogeneous ordinary differential equation

$$\begin{aligned} \begin{aligned}&\frac{d}{dt} v=-\gamma v+f{} & {} \hbox { a.e. in}\ (0,T), \\&v=g{} & {} \hbox { on}\ \{ t =0 \}. \end{aligned} \end{aligned}$$

(3.1)

Further, the solution v satisfies the following estimate

$$\begin{aligned} \Vert v\Vert _{L^{\infty }(0,T;\mathbb {R})}^{2}+\gamma ||v\Vert _{L^{2}(0,T;\mathbb {R})}^{2}+\Vert \partial _{t}v\Vert _{L^{2}(0,T;\mathbb {R})}^{2}\le C\left( |g|+\Vert f\Vert _{L^{2}(0,T;\mathbb {R})}^{2}\right) , \end{aligned}$$

where the positive constant C depends solely on T.

We now prove a statement in a similar spirit of the previous lemma, but for the abstract singular neural field problem.

Theorem 3.2

Under Hypothesis 2.1, for any $\gamma >0$, $v_0 \in L^{2}(\Omega )$ and $G\in L^{2}(\Omega _T)$, there exists a unique solution

$$\begin{aligned} v \in L^2(\Omega _T) \cap H^{1}(0,T;L^2(\Omega )) \hookrightarrow C([0,T];L^2(\Omega )), \end{aligned}$$

(3.2)

to the problem

$$\begin{aligned} \begin{aligned}&\partial _t v = -\gamma v + F(v) + G{} & {} \text {a.e. in }\Omega _T, \\&v = v_0{} & {} \text {on }\Omega \times \{t=0\}. \end{aligned} \end{aligned}$$

(3.3)

such that

$$\begin{aligned} \Vert v\Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2}+\gamma \Vert v\Vert _{L^{2}(\Omega _{T})}^{2}+\Vert \partial _{t}v\Vert _{L^{2}(\Omega _{T})}^{2}\le C_{0}^{\prime } \end{aligned}$$

for some constant $C_{0}^{\prime }$ which depends only on $\Vert v_{0}\Vert _{L^{2}(\Omega )},$ $\Vert G\Vert _{L^{2}(\Omega _{T})}$ and $K_{F}$ defined in Lemma 2.2

Proof

In order to prove the solvability of problem (3.3), we use Banach’s fixed point theorem (Evans 2022). For some constants $\tau \in [0,T]$, and $M_\tau > 0$, let us define the set

$$\begin{aligned} {\mathcal {B}}_{\tau }=\{u\in L^{\infty }(0,\tau ;L^{2}(\Omega )) :\Vert u\Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))}\le M_\tau \} \end{aligned}$$

(3.4)

and consider the operator

$$\begin{aligned} P :L^{\infty }(0,\tau ;L^{2}(\Omega ))\rightarrow L^{\infty }(0,\tau ;L^{2}(\Omega )), \qquad u \mapsto v, \end{aligned}$$

where $v=v(x,\xi ,t)$ is the solution of the ODE problem (3.1) with the right-hand side f and the initial data g, defined by

$$\begin{aligned}{} & {} f(x,\xi ,t)=G(x,\xi ,t)+F(u)(x,\xi ,t), \nonumber \\{} & {} g(x,\xi )=v_{0}(x,\xi )\quad \text {for }(x,\xi ,t)\in \Omega _{\tau }. \end{aligned}$$

(3.5)

It is important to stress that the ODE so defined has solutions $t \mapsto v(x,\xi ,t)$ in which $(x,\xi )$ are mere parameters, and in which u is an input. Further, fixed points of P are solutions to the singular neural field Eq. (2.3).

We first show that $P :\mathcal {B}_\tau \rightarrow \mathcal {B}_\tau$, is a contraction provided $\tau$ and $M_\tau$ are suitably chosen. Pick $\tau$, $M_\tau$, $u \in \mathcal {B}_\tau$. Applying Lemma 3.1 to the ODE (3.1) with functions f and g specified in (3.5), we deduce the existence of a unique solution $v=v(x,\xi ,t)$ of (3.1) dependent on parameters $(x,\xi )\in \Omega$, such that, for almost every $(x,\xi )\in \Omega$

$$\begin{aligned} \begin{aligned} \Vert v(x,\xi )\Vert _{L^{\infty }(0,\tau )}^{2}&+\gamma \Vert v(x,\xi )\Vert _{L^{2}(0,\tau )}^{2} +\Vert \partial _{t}v(x,\xi )\Vert _{L^{2}(0,\tau )}^{2} \\&\le C\left( |v_{0}(x,\xi )|^{2}+\Vert G(x,\xi )\Vert _{L^{2}(0,\tau )}^{2}+\Vert F(u)(x,\xi )\Vert _{L^{2}(0,\tau )}^{2}\right) . \qquad \end{aligned} \end{aligned}$$

Integrating this inequality over $(x,\xi )\in \Omega$, using Lemma 2.2 we obtain

$$\begin{aligned} \begin{aligned} \Vert v\Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))}^{2} +\gamma \Vert v\Vert _{L^{2}(\Omega _{\tau })}^{2}&+\Vert \partial _{t}v\Vert _{L^{2}(\Omega _{\tau })}^{2} \\&\le C(\Vert v_{0}\Vert _{L^{2}(\Omega )}^{2} + \Vert G\Vert _{L^{2}(\Omega _{\tau })}^{2}+K^2_{F})=:C_{0}^{\prime }. \end{aligned} \end{aligned}$$

Therefore setting $M_\tau =C_{0}^{\prime }$ in (3.4), we have $P :\mathcal {B}_\tau \rightarrow \mathcal {B}_\tau$. We then show that P is a contraction for small enough $\tau$. Let $u_{1}, u_{2}\in {\mathcal {B}}_{\tau }$ and $v_{i}=P(u_{i}),$ $i=1,2,$ be solutions of (3.1) with right-hand side $f_{i}$ and initial data g defined by

$$\begin{aligned} f_{i}(x,\xi ,t)=G(x,\xi ,t)+F(u_{i})(x,\xi ,t),\qquad g(x,\xi )=v_{0}(x,\xi ), \qquad (x,\xi ,t)\in \Omega _{\tau }, \end{aligned}$$

respectively. Since $v_{1}$ and $v_{2}$ satisfy the ODE (3.1), with data defined above, the difference $z=v_{1}-v_{2}$ satisfies the inequality

$$\begin{aligned} \begin{aligned} \frac{d}{dt}|z|^{2} +\gamma |z|^{2}=(F(u_{1})&-F(u_{2}))z \\&\le \frac{1}{2\gamma }|F(u_{1})-F(u_{2})|^{2}+\frac{\gamma }{2}|z|^2 \qquad \text {for a.e. }(x,\xi ,t) \in \Omega _{\tau }. \end{aligned} \end{aligned}$$

In the last passage, we have multiplied the ODE by z, and used the inequality $2\alpha \beta = 2(\alpha \varepsilon ) (\beta /\varepsilon )\le (\alpha \varepsilon )^2 + (\beta /\varepsilon )^2$ for $\alpha ,\beta ,\varepsilon \in \mathbb {R}$, with $\gamma = 1/\varepsilon ^{2}$. We now neglect the terms proportional to $\gamma z^2$, integrate over $\Omega$ with respect to $(x,\xi )$, over $(0,\tau )$ with respect to t, and use Lemma 2.2 to obtain

$$\begin{aligned} \Vert z(t)\Vert _{L^{2}(\Omega )}^{2}{} & {} \le C_{*}\int _{0}^{t} \Vert u_1(s) - u_2(s)\Vert _{L^{2}(\Omega )}^{2}ds \\{} & {} \le C_{*}t\Vert u_1 - u_2 \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2},\quad t\in (0,\tau ), \end{aligned}$$

for some constant $C_*$. Selecting $\tau$ such that $\tau C_{*}<1,$ we obtain

$$\begin{aligned} \Vert P(u_1) - P(u_2) \Vert _{L^\infty (0,\tau ;L^2(\Omega ))} = \Vert z \Vert _{L^\infty (0,\tau ;L^2(\Omega ))} < \Vert u_1 - u_2 \Vert _{L^\infty (0,\tau ;L^2(\Omega ))}, \end{aligned}$$

hence $P :\mathcal {B}_\tau \rightarrow \mathcal {B}_\tau$ is a contraction and by Banach’s fixed point theorem admits a fixed point $v=P(v)$, that is, the solution of (3.3) on the interval $[0,\tau ]$. Studying the differential Eq. (3.3) on the time interval $[\tau ,2\tau ]$, with the condition $v_0 = v(\tau )$ one proves the existence of a solution of system (3.3) on the time interval $[\tau ,2\tau ]$. This process can be continued to extend the solution on [0, T], satisfying the estimate (3.4).

Finally, we show uniqueness of the solution. If there exist two different solutions $v_{1}$ and $v_{2},$ then the difference $v=v_{1}-v_{2}$ satisfies the problem

$$\begin{aligned} \begin{aligned}&\partial _{t}v =-\gamma v+F(v_{1})-F(v_{1}){} & {} \text {on }\Omega _{T} = \Omega \times (0,T], \\&v=0{} & {} \hbox { on}\ \Omega \times \{ t = 0 \}, \end{aligned} \end{aligned}$$

Hence, we have

$$\begin{aligned} \frac{d}{dt}|v|^{2} + \gamma |v|^{2}{} & {} = (F(v_{1})-F(v_{2}))v \le \frac{1}{2\gamma }|F(v_{1})-F(v_{2})|^{2} \\{} & {} +\frac{\gamma }{2}|v|^2, \qquad \hbox { a.e.}\ (x,\xi ,t)\in \Omega _{T}. \end{aligned}$$

Integrating and using Lemma 2.2, we deduce

$$\begin{aligned} \Vert v(t) \Vert _{L^{2}(\Omega )}^{2}\le C \int _{0}^{t} \Vert v(s)\Vert _{L^{2}(\Omega )}^{2}ds\quad \text {for }t\in (0,T), \end{aligned}$$

and Gronwall’s inequality gives $v(t)=v_{1}(t)-v_{2}(t)\equiv 0$. $\square$

4 Existence of Solution to the Regular Problem $\nu >0$

To make progress towards studying solutions of the regular problem, we present a result from the theory of parabolic equations. Consider the following auxiliary linear inhomogeneous problem for unknown functions $z(x,\xi ,t)$, with $z :\Omega \times [0,T] \rightarrow \mathbb {R}$,

$$\begin{aligned} \begin{aligned}&\partial _t z = \varepsilon \partial _x^2 z + \nu \partial _{\xi }^{2} z - \gamma z + f{} & {} \text {on }\mathbb {T}^n \times U \times (0,T], \\&\partial _\xi z = 0{} & {} \text {on }\mathbb {T}^n \times \partial U \times [0,T], \\&z = g{} & {} \hbox { on}\ \mathbb {T}^n \times U \times \{ t = 0 \}. \\ \end{aligned} \end{aligned}$$

(4.1)

The following solvability result is obtained by the classical theory presented in (Evans 2022, page 378, Theorem 3), (Mikhailov 1978, page 372, Theorem 3), and we omit its proof for brevity.

Lemma 4.1

Fix $\varepsilon , \nu , \gamma >0$, $g\in L^{2}(\Omega )$, and $f\in L^{2}(\Omega _{T})$. The weak formulation of the linear inhomogeneous, anisotropic heat Eq. (4.1),

$$\begin{aligned} \begin{aligned}&\begin{aligned} \langle \partial _{t}z,\varphi \rangle + \varepsilon (\partial _{x }z,\partial _{x}\varphi )&+ \nu (\partial _{\xi }z,\partial _{\xi }\varphi ) \\&+\gamma (z,\varphi ) = (f,\varphi ), \quad \varphi \in H^{1}(\Omega ), \end{aligned}{} & {} \hbox { a.e. in}\ (0,T), \\&z(0) = g,{} & {} \hbox { on}\ U, \end{aligned} \end{aligned}$$

(4.2)

where $\langle \psi ,\varphi \rangle$ denotes the duality pairing between $\psi \in H^{1}(\Omega )^{*}$ and $\varphi \in H^{1}(\Omega )$, admits a unique solution satisfying

$$\begin{aligned} z \in L^{2}(0,T;H^{1}(\Omega ))\cap H^{1}(0,T; H^{1}(\Omega )^{*})\hookrightarrow C([0,T];L^{2}(\Omega )), \end{aligned}$$

and the estimate

$$\begin{aligned} \begin{aligned} \Vert z\Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2}&+\varepsilon \Vert \partial _{x }z\Vert _{L^{2}(\Omega _{T})}^{2} +\nu \Vert \partial _{\xi }z\Vert _{L^{2}(\Omega _{T})}^{2} \\&+\gamma \Vert z\Vert _{L^{2}(\Omega _{T})}^{2} +\Vert \partial _{t}z\Vert _{L^{2}(0,T; H^{1}(\Omega )^{*})}^{2} \!\le \! C\left( \Vert g \Vert _{L^{2}(\Omega )}^{2}+\Vert f\Vert _{L^{2}(\Omega _{T})}^{2}\right) \end{aligned} \end{aligned}$$

where $C >0$ is a constant depending solely on T and on the size L of the interval U.

The proof of existence and uniqueness of weak solutions to (2.2) relies on the following auxiliary result, which characterises weak solutions to problem (4.1) with $\varepsilon =0$, in which the dependence on x in the forcing f and initial condition g is retained.

Lemma 4.2

For any given $\nu ,\gamma >0$, $g\in L^{2}(\Omega )$ and $f\in L^{2}(\Omega _T)$, there exists a unique v satisfying the $\varepsilon =0$ weak form of (4.1), namely

$$\begin{aligned} \begin{aligned}&\begin{aligned} \langle \partial _{t}v,\varphi \rangle + \nu (\partial _{\xi }v,\partial _{\xi }\varphi )&+ \gamma (v,\varphi ) \\&= (f,\varphi ), \quad \varphi \in L^2(\mathbb {T}^n) \times H^{1}(U), \end{aligned}{} & {} \hbox { a.e. in}\ (0,T), \\&z(0) = g{} & {} \hbox { on}\ \mathbb {T}^n \times U, \end{aligned} \end{aligned}$$

(4.3)

where $\langle \psi ,\varphi \rangle$ denotes the duality pairing between $\psi \in L^2(\mathbb {T}^n) \times \left( H^{1}(U)\right) ^{*}$ and $\varphi \in L^2(\mathbb {T}^n) \times H^{1}(U)$. In addition

$$\begin{aligned} \begin{aligned} v \in L^{2}(0,T;L^2(\mathbb {T}^n) \times H^1(U))) \cap H^{1}(0,T;L^2(\mathbb {T}^n)&\times H^1(U)^*) \\&\hookrightarrow C([0,T];L^2(\Omega )), \end{aligned} \end{aligned}$$

(4.4)

and the following estimate holds

$$\begin{aligned} \begin{aligned} \Vert v\Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2}&+\nu \Vert \partial _{\xi }v\Vert _{L^{2}(\Omega _{T})}^{2} +\gamma \Vert v\Vert _{L^{2}(\Omega _{T})}^{2} +\Vert \partial _{t}v\Vert _{L^{2}(0,T;L^2(\mathbb {T})^n \times H^{1}(U)^{*})}^{2} \\&\le C\left( \Vert g \Vert _{L^{2}(\Omega )}^{2}+\Vert f\Vert _{L^{2}(\Omega _\tau )}^{2}\right) , \end{aligned} \end{aligned}$$

(4.5)

where $C >0$ is a constant depending solely on T, and on the size L of the interval U.

Proof

For fixed $\varepsilon >0$, let $v_{\varepsilon }$ be the unique weak solution to (4.2). The statement is proved by studying the $\varepsilon \rightarrow 0$ limit of $v_\varepsilon$. In passing, we note that the constant C in the estimate of Lemma 4.1 is independent of $\varepsilon$ (and $\nu$), and that the initial condition is well defined by the embedding results presented in (Evans 2022, Theorems 2, 3, pages 302–303). Owing to the estimate in Lemma 4.1, there exists a sequence $\{ v_\varepsilon \}_{\varepsilon > 0}$ and a function v such that

$$\begin{aligned} \begin{aligned} v_\varepsilon&\rightharpoonup v{} & {} *\text {-weakly in }L^\infty (0,T;L^2(\Omega ))\text { as }\varepsilon \rightarrow 0, \\ v_\varepsilon , \varepsilon \partial _x v_\varepsilon , \partial _\xi v_\varepsilon&\rightharpoonup v_\varepsilon , 0, \partial _\xi v{} & {} \text {weakly in }L^2(\Omega _T)\text { as }\varepsilon \rightarrow 0, \\ \partial _t v_\varepsilon&\rightharpoonup \partial _t v{} & {} \text {weakly in }L^2(0,T;H^1(\Omega )^*)\text { as }\varepsilon \rightarrow 0, \end{aligned} \end{aligned}$$

respectively, and using these convergence results we obtain that the limit function v satisfies (4.3). Moreover, the weak semi-continuity property of integral and the estimate in Lemma 4.1 give that v satisfies (4.5). Finally, since v is the unique solution to (4.3), we deduce that v has the regularity (4.5). $\square$

After these preliminaries, we can address the well-posedness of the weak problem for the neural field with anisotropic diffusion, system (2.2), when $\nu > 0$.

Theorem 4.3

[Weak solution, $\nu > 0$] Assume Hypothesis 2.1, and fix $\nu >0$, $\gamma \ge 0$, $v_0 \in L^2(\Omega )$, and $G \in L^2(\Omega _T)$. There exists a unique v satisfying the regularity condition (4.4) such that

$$\begin{aligned} \begin{aligned}&\begin{aligned} \langle \partial _{t}v,\varphi \rangle +&\nu (\partial _{\xi }v,\partial _{\xi }\varphi ) +\gamma (v,\varphi ) \\&= (F(v)+G,\varphi ), \qquad \varphi \in L^{2}(\mathbb {T}^n) \times H^{1}(U), \end{aligned}{} & {} \hbox { a.e. in}\ (0,T), \\&v(0) =v_0,{} & {} \hbox { on}\ \mathbb {T}^n \times U. \end{aligned} \end{aligned}$$

(4.6)

In addition, there exists a constant $C_0$ depending on $\Vert G \Vert _{L^2(\Omega _T)}$, $\Vert v_0 \Vert _{L^2(\mathbb {T}^n \times U)}$, and on the constant $K_F$ defined in Lemma 2.2, such that

$$\begin{aligned} \Vert v\Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2} +\nu \Vert \partial _{\xi } v \Vert _{L^{2}(\Omega _{T})}^{2} +\gamma \Vert v\Vert _{L^{2}(\Omega _{T})}^{2} +\Vert \partial _{t}v\Vert _{L^{2}(0,T;L^{2}({\mathbb {T}}^{n})\times H^{1}(U)^{*})}^{2}\le C_{0}. \end{aligned}$$

Proof

The structure of this proof resembles the one of Theorem 3.2: we use a local-in-time Banach fixed point argument, which we bootstrap to extend the solution on [0, T]. Consider the set

$$\begin{aligned} \mathcal {B}_\tau = \{ u\in L^{\infty }(0,\tau ;L^{2}(\Omega )) :\Vert u\Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))}\le M_\tau \}, \end{aligned}$$

(4.7)

that is, the closed ball of radius $M_\tau$ centred at the origin in $L^{\infty }(0,\tau ;L^{2}(\Omega ))$, where $\tau \in [0,T]$ and $M_\tau$ will be specified later on. We define the operator

$$\begin{aligned} P :L^{\infty }(0,\tau ;L^{2}(\Omega ))\rightarrow L^{\infty }(0,\tau ;L^{2}(\Omega )), \qquad u \mapsto v, \end{aligned}$$

where v is the solution to the weak problem (4.3) in Lemma 4.2 with forcing and initial conditions

$$\begin{aligned} f(x,\xi ,t) = F(u)(x,\xi ,t) + G(x,\xi ,t), \qquad g(x,\xi ) = v_0(x,\xi ), \qquad (x,\xi ,t) \in \Omega _\tau . \end{aligned}$$

(4.8)

We claim the existence of $\tau \in [0,T]$ and of $M_\tau > 0$ such that $P :\mathcal {B}_\tau \rightarrow \mathcal {B}_\tau$ is a contraction which in turn, by Banach’s fixed point theorem, implies the existence on the time interval $(0,\tau )$ of a solution to (4.6), that is, a weak solution to the nonlinear neural field problem with anisotropic diffusion (2.2) with $\nu > 0$.

Firstly, we select $M_\tau$ so that $P :\mathcal {B}_\tau \rightarrow \mathcal {B}_\tau$ for any $\tau \in [0,T]$. Lemma 4.2 ensures P is injective. The estimate (4.5), of which we neglect the second, third, and fourth term on the left-hand side, guarantees that P is indeed on $L^{\infty }(0,\tau ;L^{2}(\Omega ))$ to itself. The choice

$$\begin{aligned} M_\tau = C \bigg [ \Vert v_0 \Vert ^2_{L^2(\Omega )} + \Vert G \Vert ^2_{L^2(\Omega _\tau )} + \tau ^2 K_{F}^2 \bigg ], \end{aligned}$$

guarantees also that $\Vert P(u) \Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))} \le M_\tau$, hence $P :\mathcal {B}_\tau \rightarrow \mathcal {B}_\tau$ for any $\tau \in [0,T]$.

Secondly, we show that P is a contraction, that is, we prove that, for sufficiently small $\tau$, we can find a constant $K_* \in [0,1)$ such that

$$\begin{aligned} \Vert P(u_1) - P(u_2) \Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))} \le K_* \Vert u_1 - u_2 \Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))} \qquad u_1, u_2 \in \mathcal {B}_\tau \end{aligned}$$

To this end, fix $u_1, u_2 \in \mathcal {B}_\tau$, and let $v_1=P(u_1)$, $v_2=P(u_2)$, that is, the unique solutions to the weak problem in Lemma 4.2, with

$$\begin{aligned} f_i(x,\xi ,t) = F(u_i)(x,\xi ,t) + G(x,\xi ,t), \qquad i=1,2, \qquad (x,\xi ,t) \in \Omega _\tau , \end{aligned}$$

while keeping all other data constant. The function $z = v_1 - v_2$ satisfies, for any test function $\varphi \in L^2(\mathbb {T}^n) \times H^{1}(U)$

$$\begin{aligned} \begin{aligned} \langle \partial _{t} z,\varphi \rangle + \nu (\partial _{\xi } z,\partial _{\xi }\varphi )+\gamma (z,\varphi )&= (F(u_1)-F(u_2),\varphi ),{} & {} \hbox { a.e.}\ (0,\tau ), \\ z(0)&=0,{} & {} \hbox { on}\ \Omega . \end{aligned} \end{aligned}$$

We choose $\phi = z(t)$, recall that

$$\begin{aligned} \langle \partial _t z, z \rangle = \frac{d}{dt} \Vert z \Vert ^2_{L^2(\Omega )} \qquad \hbox { a.e. in}\ (0,\tau ), \qquad z \in C([0,\tau ];L^2(\Omega )), \end{aligned}$$

(4.9)

and by the result in ( Málek et al. (1996), page 35) we have

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\Vert z \Vert ^2_{L^2(\Omega )}&+ \nu \Vert \partial _\xi z \Vert ^2_{L^2(\Omega )} + \gamma \Vert z \Vert ^2_{L^2(\Omega )} \\&\le \frac{1}{2\gamma } \Vert F(u_1) - F(u_2)\Vert ^2_{L^2(\Omega )} + \frac{\gamma }{2} \Vert z \Vert ^2_{L^2(\Omega )} \quad \text {a.e. in }(0,\tau ). \\ \end{aligned} \end{aligned}$$

Neglecting the second and third terms on the left-hand side, and using that F is Lipschitz function, we arrive at

$$\begin{aligned} \begin{aligned} \Vert v_1(t) - v_2(t) \Vert ^2_{L^2(\Omega )}&\le C_* \int _0^t \Vert u_1(s) - u_2(s) \Vert ^2_{L^2(\Omega )}\, ds \\&\le C_* t \Vert u_1 - u_2 \Vert ^2_{L^\infty (0,\tau ;L^2(\Omega ))} \qquad \text {for all }t \in (0,\tau ), \end{aligned} \end{aligned}$$

for some positive constant $C_*$ depending on $K_F$, as defined in Lemma 2.2. Therefore, picking $\tau < 1/C_*$ we obtain

$$\begin{aligned} \Vert P(u_1) - P(u_2) \Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))} < \Vert u_1 - u_2 \Vert _{L^{\infty }(0,\tau ;L^{2}(\Omega ))} \qquad u_1, u_2 \in \mathcal {B}_\tau . \end{aligned}$$

We have thus proven the existence of a unique solution to (4.6) on $[0,\tau ]$. The function v is continuous on the time variable with values in $L^2(\Omega )$ by (4.4), and can be extended to a continuous function on $[0,2\tau ]$ by repeating the steps above on the initial-value problem (4.6) posed on $(\tau ,2\tau )$ with initial condition $v(x,\xi ,\tau )$. Iterating this process, we find a continuous solution on the whole [0, T]. It remains to show that the resulting solution v is unique. If two solutions $v_1, v_2$ then $v=v_1-v_2$ satisfies the equality (4.6) with $F(v)=F(v_1) - F(v_2)$, initial condition $v_0 =0$, and $G=0$. Choosing $\phi =v(t)$ in this equality leads to

$$\begin{aligned} \begin{aligned} \frac{d}{dt} \Vert v \Vert _{L^{2}(\Omega )}^{2}&+\nu \Vert \partial _{\xi }v\Vert _{L^{2}(\Omega )}^{2} +\gamma \Vert v\Vert _{L^{2}(\Omega )}^{2}=(F(v_{1})-F(v_{2}),v) \\&\le \frac{1}{2\gamma } \Vert F(v_1) - F(v_2)\Vert ^2_{L^2(\Omega )} + \frac{\gamma }{2} \Vert v \Vert ^2_{L^2(\Omega )} \quad \hbox { a.e. in}\ (0,T). \\ \end{aligned} \end{aligned}$$

Then, integrating this inequality with respect to the time variable over the interval (0, t) and using Lemma 2.2, we obtain

$$\begin{aligned} ||v(t)\Vert _{L^{2}(\Omega )}^{2}\le {\tilde{C}}\int _{0}^{t}||v(s)\Vert _{L^{2}(\Omega )}^{2}ds\quad \text {for a.e. }t\in (0,T). \end{aligned}$$

By Gronwall’s inequality, we conclude $v(t)=v_{1}(t)-v_{2}(t)\equiv 0$.

Finally, setting (4.8) on the whole $\Omega _T$, using the estimate (4.5), and Lemma 2.2 we derive the embedding presented in the theorem statement, with

$$\begin{aligned} C_0 = C \bigl ( \Vert v_0 \Vert _{L^2(\Omega )} + \Vert G \Vert _{L^2(\Omega _T)} + K_{F}\bigr ). \end{aligned}$$

$\square$

5 Continuous Dependence on the Diffusion Parameter $\nu$

Now that we have studied the well-posedness of the regular and singular problem, we can address the dependence of solutions on the diffusion parameter $\nu$. We prove continuous dependence on $\nu$ by expanding both solutions in a suitable basis of $L^2(\Omega )$, and bounding their difference.

Theorem 5.1

Assume Hypothesis 2.1, and fix $\gamma \ge 0$, $v_0 \in L^2(\Omega )$, and $G \in L^2(\Omega _T)$. The solutions v(t) and $v_{\nu }(t)$ of the singular problem (3.3), and of the regular problem (4.6), respectively, satisfy

$$\begin{aligned} \Vert v - v_\nu \Vert _{C([0,T];L^{2}(\Omega ))} \rightarrow 0 \qquad \text {as }\nu \rightarrow 0. \end{aligned}$$

(5.1)

Proof

Using the integral operator F in (2.1), throughout the proof we set

$$\begin{aligned} N = G + F(v), \qquad N_\nu = G + F(v_\nu ), \qquad \text {a.e in} \Omega _T. \end{aligned}$$

(5.2)

By Theorems 3.2 and 4.3, we have $v,v_{\nu }\in L^{2}(\Omega _{T})$, uniformly on $\nu >0$. The main hypothesis $v_0 \in L^2(\Omega )$ and Lemma 2.2 give

$$\begin{aligned} \Vert N \Vert ^2_{L^2(\Omega _T)}, \Vert N_{\nu } \Vert ^2_{L^2(\Omega _T)} < \infty , \end{aligned}$$

which, combined with Fubini’s theorem, imply

$$\begin{aligned} N(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}},{\hspace{1.111pt}\cdot \hspace{1.111pt}}), N_{\nu }(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}},{\hspace{1.111pt}\cdot \hspace{1.111pt}}) \in L^2(U_T), \qquad v_0(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}}) \in L^2(U) \qquad \text {for a.e. }x \in \mathbb {T}^n. \end{aligned}$$

(5.3)

To demonstrate (5.1), it is enough to show that for any $\varepsilon >0$ there exists a function $\nu :\mathbb {R}_{ \ge 0} \rightarrow \mathbb {R}_{ \ge 0}$ such that

$$\begin{aligned} \Vert v - v_{\nu } \Vert ^2_{C([0,T]; L^2(\Omega ))}< \varepsilon \qquad \text {for all }\nu < \nu (\varepsilon ). \end{aligned}$$

The proof proceeds in steps, and leads to the norm estimate for $v(t)-v_\nu (t)$ through various calculations involving point-wise estimates and evaluations for fixed $x \in \mathbb {T}^n$. We shall sometimes omit the dependence on x, to simplify the notation.

Step 1: eigenvalues and eigenfunctions. Since $v_\nu (t)$ satisfies the system (4.6), then, by the classical results (Mikhailov 1978), for each $x \in \mathbb {T}^n$ we can expand $v_{\nu }$ as a linear combination of eigenfunctions of the following spectral problem associated to (4.6),

$$\begin{aligned} \begin{aligned} \nu \psi '' - \gamma \psi = \lambda \psi , \quad \text {on }U, \qquad \psi = 0, \quad \text {on }\partial U. \end{aligned} \end{aligned}$$

(5.4)

The eigenpairs $\{ (\lambda _k,\psi _k) \}_{k \ge 0}$ of this problem are known

$$\begin{aligned} \begin{aligned}&\psi _0(\xi ) = \sqrt{\frac{1}{L}},{} & {} \lambda _0 = -\gamma ,{} & {} \\&\psi _k(\xi ) = \sqrt{\frac{2}{L}} \cos \biggl ( \frac{k \pi \xi }{L}\biggr ),{} & {} \lambda _k = -\gamma -\nu \biggl ( \frac{k \pi \xi }{L}\biggr ),{} & {} k \ge 1, \\ \end{aligned} \end{aligned}$$

and the set $\{ \psi _k \}_{k \ge 0}$ is an orthogonal basis in $L^2(U)$ with respect to the standard inner product and norm

$$\begin{aligned} (u,z)_{L^2(U)}=\int _{0}^{L}u(\xi )z(\xi )d\xi , \qquad \Vert u \Vert ^2_{L^{2}(U)}=(u,u)_{L^2(U)}, \end{aligned}$$

respectively. Moreover, $\{\psi _{k}\}_{k\ge 0}$ is also the orthogonal basis in the space $H^{1}(U)$ with respect to inner product and norm

$$\begin{aligned} (u,z)_{H_1(U)}= \int _{0}^{L}\big [ \gamma u(\xi )z(\xi )+ u'(\xi )z'(\xi )\big ] d\xi , \qquad \Vert u \Vert ^2_{H^1(U)}=(u,u)_{H^1(U)}. \end{aligned}$$

Step 2: expansion of $v_\nu$. We now seek to expand $v_{\nu }(x,\xi ,t)$ in the form

$$\begin{aligned} v_{\nu }(x,\xi ,t)=\sum _{k=0}^{\infty } g_{k}(x,t)\psi _{k}(\xi ), \end{aligned}$$

(5.5)

and we determine coefficients $g_k$ so that $v_\nu$ satisfies (4.6) in the sense of distributions. By the initial condition in (4.6), it follows that

$$\begin{aligned} v_{\nu }(x,\xi ,0)=\sum _{k=0}^{\infty }g_{k}(x,0)\psi _{k}(\xi )=v_{0}(x,\xi ). \end{aligned}$$

(5.6)

On the other hand, applying Parseval’s equality and using (5.3), we have

$$\begin{aligned} \begin{aligned}&v_{0}(x,\xi ) = \sum _{k=0}^{\infty }v_{0,k}(x)\psi _{k}(\xi ),{} & {} v_{0,k}(x)=\bigl (v_{0}(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}}),\psi _{k}\bigr )_{L^2(U)}, \\&N_{\nu }(x,\xi ,t) =\sum _{k=0}^{\infty }N_{\nu ,k}(x,t)\psi _{k}(\xi ),{} & {} N_{\nu ,k}(x,t)=\bigl (N_{\nu }(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}},t),\psi _{k}\bigr )_{L^2(U)}, \end{aligned} \end{aligned}$$

(5.7)

which imply

$$\begin{aligned} \begin{aligned}&\Vert v_{0}(x)\Vert _{L^{2}(U)}^{2} = \sum _{k=0}^{\infty }v_{0,k}^{2}(x)< \infty ,{} & {} \text {a.e. }x\in {\mathbb {T}}^{n}, \\&\Vert N_{\nu }(x,t)\Vert _{L^{2}(U)}^{2} = \sum _{k=0}^{\infty }N_{\nu ,k}^{2}(x,t) < \infty ,{} & {} \hbox { a.e.}\ (x,t)\in {\mathbb {T}}^{n}\times [0,T], \end{aligned} \end{aligned}$$

(5.8)

and, by the monotone convergence theorem, we obtain

$$\begin{aligned} \Vert N_{\nu }(x)\Vert _{L^{2}(U_{T})}^{2} =\sum _{k=0}^{\infty }\int _{0}^{T}N_{\nu ,k}^{2}(x,t)dt <\infty , \qquad \text {a.e. }x\in {\mathbb {T}}^{n}. \end{aligned}$$

(5.9)

Integrating (5.8) and (5.9) over $x\in {\mathbb {T}}^{n}$ and applying again the monotone convergence theorem, we have

$$\begin{aligned} \begin{aligned}&\Vert v_{0}\Vert _{L^{2}(\Omega )}^{2} =\sum _{k=0}^{\infty } \hat{v}_{0,k}^{2}< \infty ,{} & {} \hat{v}_{0,k}^{2}=\int _{{\mathbb {T}}^{n}} v_{0,k}^{2}(x)dx, \\&\Vert N_{\nu } \Vert _{L^{2}(\Omega _{T})}^{2} =\sum _{k=0}^{\infty }\int _{0}^{T}{\hat{N}}_{\nu ,k}^{2}(t)dt <\infty ,{} & {} {\hat{N}}_{\nu ,k}^{2}(t)=\int _{{\mathbb {T}}^{n}}N_{\nu ,k}^{2}(x,t)dx. \end{aligned} \end{aligned}$$

(5.10)

Substituting the expansions (5.5)-(5.7) into system (4.6), we conclude that for each $k \in \mathbb {Z}_{ \ge 0}$, the coefficients $g_{k}(t)$ satisfy the following initial-value problem

$$\begin{aligned} \begin{aligned}&\partial _t g_k(x,t) = \lambda _k g_k(x,t) + N_\nu (x,t),{} & {} (x,t) \in \mathbb {T}^n \times [0,T], \\&g_k(x,0) = v_{0,k}(x),{} & {} x \in \mathbb {T}^n, \end{aligned} \end{aligned}$$

(5.11)

and are therefore given by

$$\begin{aligned} g_{k}(x,t)=v_{0,k}(x) \exp \bigl (\lambda _{k}t\bigr ) + \int _{0}^{t}N_{\nu ,k}(x,s)\exp \bigl ( \lambda _{k}(t-s)\bigr ) ds, \end{aligned}$$

(5.12)

such that $g_{k}(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}}) \in H^{1}(0,T)\subset C([0,T])$ for a.e. $x\in {\mathbb {T}}^{n}$. This, together with (5.5), completes the expansion of $v_\nu$.

$$\begin{aligned} v_\nu (x,\xi ,t) = \sum _{k=0}^{\infty } \psi _{k}(\xi ) \Bigl [v_{0,k}(x)\exp \left( \lambda _k t\right) + \int _{0}^{t}N_{\nu ,k}(x,s)\exp \left( \lambda _k(t-s)\right) ds \Bigr ]. \end{aligned}$$

(5.13)

Step 3: expansion of v. The next step is to expand the solution v of the singular neural field system (3.3) in terms of the eigenfunctions $\psi _{k}$. The solution of the problem (3.3) can be written in the form

$$\begin{aligned} v(x,\xi ,t) = v_{0}(x,\xi ) e^{-\gamma t} +\int _{0}^{t} N(x,\xi ,s)e^{-\gamma (t-s)} ds. \end{aligned}$$

(5.14)

By the regularity (5.3), we have

$$\begin{aligned} N(x,\xi ,t)=\sum _{k=0}^{\infty }N_{k}(x,t)\psi _{k}(\xi ), \qquad N_{k}(x,t)=\bigl (N(x,{\hspace{1.111pt}\cdot \hspace{1.111pt}},t),\psi _{k}\bigr )_{L^2(U)}, \end{aligned}$$

(5.15)

and by Parseval’s equality and the monotone convergence theorem

$$\begin{aligned} \begin{aligned}&\Vert N(x,t)\Vert _{L^{2}(U)}^{2} = \sum _{k=0}^{\infty } N_{k}^{2}(x,t)<\infty ,{} & {} \hbox { a.e.}\ (x,t)\in {\mathbb {T}}^n \times [0,T], \\&\Vert N(x) \Vert _{L^{2}(U_{T})}^{2} = \sum _{k=0}^{\infty }\int _{0}^{T}N_{k}^{2}(x,t) dt,{} & {} \hbox { a.e.}\ x\in {\mathbb {T}}^n, \\&\Vert N \Vert _{L^{2}(\Omega _{T})}^{2} = \sum _{k=0}^{\infty }\int _{0}^{T} {\hat{N}}_{k}^{2}(t)dt < \infty ,{} & {} {\hat{N}}_{k}^{2}(t)=\int _{{\mathbb {T}}^{n}}N_{k}^{2}(x,t) dx \end{aligned} \end{aligned}$$

(5.16)

Substituting (5.7) and (5.15) into the right-hand side of (5.14), we obtain

$$\begin{aligned} v(x,\xi ,t) = \sum _{k=0}^{\infty } \psi _{k}(\xi ) \biggl [v_{0,k}(x)e^{-\gamma t} +\int _{0}^{t}N_{k}(x,s) e^{-\gamma (t-s)} ds \biggr ]. \end{aligned}$$

(5.17)

Step 4: expanding $v-v_\nu$ in sums over functions $A_k$ and $B_k$. Using the previous two steps, we will now derive a priori estimate for the difference of v and $v_{\nu }$ in the norm of the Banach space $L^{2}(U)$. Henceforth, we write $-\mu _k(\nu ) = -\nu (k\pi /L)^2 = \lambda _k(\nu ,\gamma ) + \gamma$, where $\lambda _k$ are the eigenvalues (5.4), and omit the dependence on $\nu$ when possible. Combining these definitions with (5.17), (5.5), and (5.12), we obtain

$$\begin{aligned} \begin{aligned} v(x,\xi ,t)-v_{\nu }(x,\xi ,t)&= \sum _{k=0}^{\infty }\psi _{k}(\xi ) \biggl [ v_{0,k}(x) ( 1- e^{-\mu _k t}) e^{-\gamma t} \\&+ \int _{0}^{t}\bigl ( N_{k}(x,s)-N_{\nu ,k}(x,s) e^{-\mu _k (t-s)}\bigr ) e^{-\gamma (t-s)} ds \biggr ]. \end{aligned} \end{aligned}$$

Since $\{\psi _{k}\}_{k>0}$ is the orthonormal basis in $L^{2}(U)$, then taking the $L^{2}(U)$-norm of the previous expressions we obtain

$$\begin{aligned} \Vert v(x,t)-v_{\nu }(x,t)\Vert _{L^{2}(\Omega )}^{2} =\sum _{k=0}^{\infty }(A_{k}+B_{k})^{2} \le 2\sum _{k=0}^{\infty }A_{k}^{2}+2\sum _{k=0}^{\infty }B_{k}^{2}, \end{aligned}$$

(5.18)

where

$$\begin{aligned} \begin{aligned}&A_{k}(x,t) = v_{0,k}(x)( 1- e^{-\mu _k t}) e^{-\gamma t}, \\&B_{k}(x,t) = \int _{0}^{t}\bigl [ N_{k}(x,s)-N_{\nu ,k}(x,s) e^{-\mu _k (t-s)}\bigr ] e^{-\gamma (t-s)} ds, \end{aligned} \end{aligned}$$

(5.19)

and a further integration over $x \in \mathbb {T}^n$ gives

$$\begin{aligned} \Vert v(t)-v_{\nu }(t)\Vert _{L^{2}(\Omega )}^{2} \le 2\sum _{k=0}^{\infty }\int _{{\mathbb {T}}^{n}} A_{k}^{2}dx +2\sum _{k=0}^{\infty }\int _{{\mathbb {T}}^{n}}B_{k}^{2}dx. \end{aligned}$$

(5.20)

Step 5: bounding the sum in $A^2_k$ in the expansion (5.20). By the first equality in (5.10), we have that for any $\epsilon >0$ there exists $k_{1}(\epsilon )>0,$ such that

$$\begin{aligned} \sum _{k=k_{1}}^{\infty }\hat{v}_{0,k}^{2}<\frac{\epsilon }{4D}, \end{aligned}$$

where the constant $D>0$ will be chosen below, at the very end of the proof. For all $k \in \mathbb {Z}_{\ge 0}$ and $t\in [0,T]$, it holds $0 \le ( 1- e^{-\mu _k t}) e^{-\gamma t} \le 1$, hence we estimate

$$\begin{aligned} \sum _{k=k_{1}}^{\infty }\int _{{\mathbb {T}}^{n}}A_{k}^{2}(x,t)dx<\frac{\epsilon }{4D}, \qquad t\in [0,T]. \end{aligned}$$

(5.21)

Moreover, for almost any $x \in \mathbb {T}^n$, and all $t \in [0,T]$ we have

$$\begin{aligned} \sum _{k=0}^{k_{1}-1}A_{k}^{2}(x,t) \le \sum _{k=0}^{k_{1}-1}|v_{0,k}(x)|^{2} ( 1- e^{-\mu _{k_1} t})^2 e^{-2\gamma t} \le \Vert v_{0}(x)\Vert _{L^{2}(U)}^{2}( 1- e^{-\mu _{k_1} T})^2, \end{aligned}$$

whence

$$\begin{aligned} \sum _{k=0}^{k_{1}-1}\int _{{\mathbb {T}}^{n}}A_{k}^{2}(x,t)dx \le \Vert v_{0}\Vert _{L^{2}(\Omega )}^{2}( 1- e^{-\mu _{k_1} T})^2, \qquad t \in [0,T] \end{aligned}$$

(5.22)

Recalling the dependence of $\mu _k(\nu )$, and using the continuity and monotonicity of the exponential function, we obtain that for any $\varepsilon > 0$, there exists $\nu _1(\varepsilon )$ such that

$$\begin{aligned} ( 1- e^{-\mu _{k_1}(\nu ) T})^2 = ( 1- e^{-\nu (k_1\pi /L)^2 T})^2< \frac{\varepsilon }{4 \Vert v_0 \Vert ^2_{L^2(\Omega )} D} \qquad \text {for all }\nu < \nu _1(\varepsilon ). \end{aligned}$$

Using (5.21) and (5.22), we deduce that for any $\epsilon >0$ there exists $\nu _{1}(\epsilon )$ such that

$$\begin{aligned} \sum _{k=0}^{\infty }\int _{{\mathbb {T}}^{n}}A_{k}^{2}(x,t)dx< \frac{\epsilon }{2D}, \qquad \text {for all }\nu <\nu _{1}(\epsilon )\text { and }t\in [0,T], \end{aligned}$$

(5.23)

that is, we have found a bound for the sum in the terms $A_k$ in (5.20).

Step 6: bounding the sum in $B^2_k$ in the expansion (5.20). With this goal, we rewrite the $B_k$ in (5.19) in the form

$$\begin{aligned} \begin{aligned} B_{k}(x,t)&= \int _{0}^{t} N_{k}(x,s)(1- e^{-\mu _k (t-s)}) e^{-\gamma (t-s)} ds \\&+ \int _{0}^{t} \bigl [ N_{k}(x,s)-N_{\nu ,k}(x,s)\bigr ] e^{-\mu _k (t-s)} e^{-\gamma (t-s)} ds, \end{aligned} \end{aligned}$$

Recalling that $(\alpha +\beta )^{2}\le 2\alpha ^{2}+2\beta ^{2}$, it follows that

$$\begin{aligned} B^2_{k}(x,t){} & {} = 2T(1- e^{-\mu _k T})^2 \int _{0}^{t} N^2_{k}(x,s) ds \nonumber \\{} & {} \quad + 2T \int _{0}^{t} \bigl [ N_{k}(x,s)-N_{\nu ,k}(x,s)\bigr ] ds. \end{aligned}$$

(5.24)

From the third inequality in (5.16), we deduce that, for any $\epsilon >0$, there exists an integer $k_2 = k_{2}(\epsilon )$ such that

$$\begin{aligned} \sum _{k=k_{2}}^{\infty } \int _{0}^{T} {\hat{N}}_{k}^{2}(s)ds < \frac{\epsilon }{4D}(2T)^{-1}, \end{aligned}$$

hence, integrating (5.24) over $x\in {\mathbb {T}}^{n}$ we obtain

$$\begin{aligned} \begin{aligned} \sum _{k=0}^{\infty }\int _{{\mathbb {T}}^{n}}B_{k}^{2}(x,t)dx&\le 2T \Biggl ( \sum _{k=k_{2}}^{\infty } \int _{0}^{T}\hat{N}_{k}^{2}(s)ds \Biggr ) \\&+2T(1- e^{-\mu _{k_2} T})^2 \Biggl ( \sum _{k=0}^{k_{2}-1}\int _{0}^{T}\hat{N}_{k}^{2}(s)ds\Biggr ) \\&+ 2T \int _{0}^{t} \sum _{k=1}^{\infty } \int _{\mathbb {T}^n} \bigl | N_{k}(x,s)-N_{\nu ,k}(x,s) \big |^2 dx ds. \end{aligned} \end{aligned}$$

Using the definition of N in (5.2), and the Lipschitzianity of the function F, we estimate

$$\begin{aligned} \begin{aligned} \sum _{k=0}^{\infty }\int _{{\mathbb {T}}^{n}}B_{k}^{2}(x,t)dx \le \frac{\varepsilon }{4D}&+ 2T (1- e^{-\mu _{k_2(\varepsilon )}(\nu ) T})^2 \Vert G + F(v) \Vert ^2_{L^2(\Omega _T)} \\&+ 2T K_F \int _{0}^{t} \Vert u(s)- u_\nu (s) \Vert ^2_{L^2(\Omega )} ds. \end{aligned} \end{aligned}$$

We choose $\nu _2(\varepsilon )$ such that

$$\begin{aligned} (1- e^{-\mu _{k_2(\varepsilon )}(\nu ) T})^2 \le \frac{\epsilon }{4D} \left( 2T\Vert G + F(v)\Vert _{L^{2}(\Omega _{T})}^{2}\right) ^{-1}, \qquad \hbox { for all}\ \nu < \nu _{2}(\epsilon ), \end{aligned}$$

and we conclude

$$\begin{aligned} \begin{aligned} \sum _{k=0}^{\infty }\int _{{\mathbb {T}}^{n}}B_{k}^{2}(x,t)dx \le \frac{\varepsilon }{2D}&+ 2T K_F \int _{0}^{t} \Vert u(s)- u_\nu (s) \Vert ^2_{L^2(\Omega )} ds. \end{aligned} \end{aligned}$$

(5.25)

Step 7: final estimate. We set $\nu (\epsilon )=\min (\nu _{1}(\epsilon ),\nu _{2}(\epsilon ))$, and combine (5.18), (5.23), and (5.25) to obtain

$$\begin{aligned} \Vert v(t)-v_{\nu }(t)\Vert _{L^{2}(\Omega )}^{2} \le \frac{\epsilon }{D} +2T K_F \int _{0}^{t}\Vert v(s)-v_{\nu }(s)\Vert _{L^{2}(\Omega )}^{2}ds, \qquad \text {for all }\nu <\nu (\epsilon ). \end{aligned}$$

Applying Gronwall’s inequality, recalling that D is a so far unspecified constant, and picking $D=e^{2T^2K_F}$, we find

$$\begin{aligned} \Vert v(t)-v_{\nu }(t)\Vert ^2_{L^{2}(\Omega )} \le \frac{e^{2T^2K_F}}{D}\epsilon = \varepsilon ,\qquad \hbox { for all } \nu <\nu (\epsilon ) \hbox { and } t\in [0,T], \end{aligned}$$

hence

$$\begin{aligned} \Vert v-v_{\nu }\Vert _{C([0,T];L^{2}(\Omega ))} \rightarrow 0 \qquad \hbox { as}\ \nu \rightarrow 0, \end{aligned}$$

and the proof is complete. $\square$

6 Convergence Rate of the Sequence $v_\nu - v$ to 0

The statement below shows that if, in addition to the hypotheses of Theorem 5.1, one adds regularity assumptions to the initial data and external input, then it is possible to prove $v_\nu - v$ is $O(\nu ^{1/2})$.

Theorem 6.1

Assume Hypothesis 2.1, and fix $\gamma \ge 0$, $v_0 \in L^2(\Omega )$, and $G \in L^2(\Omega _T)$. If, in addition, $\partial _\xi v_0 \in L^2(\Omega )$ and $\partial _\xi G \in L^2(\Omega _T)$, then the solutions v and $v_{\nu }$ to (3.3) and (4.6), respectively, satisfy

$$\begin{aligned} \Vert v_{\nu }-v\Vert _{L^{\infty }(0,T;L^{2}(\Omega ))} \le C e^{DT} \nu ^{1/2}, \end{aligned}$$

(6.1)

where positive constants C and D depend on $\gamma$, the constant $K_F$ in Leema 2.2, and the norms $\Vert v_{0}\Vert _{L^{2}(\Omega )}$, $\Vert \partial _{\xi } v_{0} \Vert _{L^{2}(\Omega )}$, $\Vert G \Vert _{L^{2}(\Omega _{T})}$, and $\Vert \partial _{\xi }G\Vert _{L^{2}(\Omega _{T})}$.

Proof

We initially find an estimate for $v - v_{\nu }$ in a different way to what was done in the proof of Theorem 5.1. This new derivation will reveal that estimate (6.1) is possible if we gain control over $\partial _\xi v$, which we then proceed to bound.

The difference $w=v_{\nu }-v$ satisfies, for any test function

$$\begin{aligned} \begin{aligned}&\begin{aligned} \langle \partial _{t}w,\varphi \rangle&+ \nu (\partial _{\xi }v_{\nu },\partial _{\xi }\varphi ) \\&+\gamma (w,\varphi ) = (F(v_{\nu })-F(v),\varphi ), \quad \phi , \partial _\xi \phi \in L^2(\Omega ), \end{aligned}{} & {} \hbox { a.e. in}\ (0,T) \\&w =0{} & {} \hbox { on}\ \Omega \times \{ t=0 \}. \end{aligned} \end{aligned}$$

Choosing the test function $\varphi =w$ gives, for almost any in $t \in (0,T)$:

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\Vert w \Vert _{L^{2}(\Omega )}^{2}&+ \nu \Vert \partial _\xi v_\nu \Vert ^2_{L^2(\Omega )} + \gamma \Vert w \Vert _{L^{2}(\Omega )}^{2} = \bigl (F(v_{\nu })-F(v),w\bigr ) + \nu (\partial _{\xi }v_{\nu },\partial _{\xi }v) \\&\le \frac{2}{\gamma } \Vert F(v_{\nu })-F(v)\Vert _{L^{2}(\Omega )}^{2} +\frac{\gamma }{2} \Vert w\Vert _{L^{2}(\Omega )}^{2} \\&+\frac{\nu }{2}\Vert \partial _{\xi }v_{\nu }\Vert _{L^{2}(\Omega )}^{2} +\frac{\nu }{2}\Vert \partial _{\xi } v\Vert _{L^{2}(\Omega )}^{2}. \end{aligned} \end{aligned}$$

Disregarding the terms in $\Vert \partial _\xi v_\nu \Vert _{L^2(\Omega )}$, integrating with respect to the time variable over the interval (0, t), and using the Lipschitzianity of F from Lemma 2.2, we obtain

$$\begin{aligned}{} & {} \Vert w(t) \Vert _{L^{2}(\Omega )}^{2} + \int _{0}^{t}\frac{\gamma }{2} \Vert w(s) \Vert _{L^{2}(\Omega )}^{2}ds \nonumber \\{} & {} \quad \le C_1 \int _{0}^{t} \Vert w(s)\Vert _{L^{2}(\Omega )}^{2}ds +\frac{\nu }{2} \int _{0}^{t} \Vert \partial _{\xi }v(s)\Vert _{L^{2}(\Omega )}^{2} ds, \end{aligned}$$

(6.2)

for all $t \in (0,T)$, where $C_1= 2K_F/\gamma$. Inspecting the previous inequality, we note that a bound on $\Vert \partial _\xi v(t) \Vert _{L^2(\Omega )}$ is necessary to find an estimate for $\Vert w(t) \Vert ^2_{L^2(\Omega )}$. We put the inequality above aside, and we devote the rest of the proof to finding the bound on $\Vert \partial _\xi v(t) \Vert _{L^2(\Omega )}$.

We begin by extending the singular neural field solution v in the spatial variable $\xi$. For fixed $\delta >0$, we set

$$\begin{aligned} {\hat{U}} = (-\delta ,L+\delta ) \supset U, \qquad {\hat{\Omega }} = \mathbb {T}^n \times {\hat{U}} \supset \Omega , \qquad {\hat{\Omega }}_T = {\hat{\Omega }} \times [0,T] \supset \Omega _T. \end{aligned}$$

Following the approach of (Evans 2022, Theorem 1, p. 268), we deduce that for any $\delta >0$ there exist functions $\hat{v}_{0}$ and $\hat{G}$ such that

$$\begin{aligned} \begin{aligned}&{\hat{v}}_0 \in L^2({\hat{\Omega }}),{} & {} \partial _\xi {\hat{v}}_0 \in L^2({\hat{\Omega }}),{} & {} {\hat{v}}_0 = v_0{} & {} \hbox { in}\ \Omega , \\&{\hat{G}} \in L^2({\hat{\Omega }}_T),{} & {} \partial _\xi {\hat{G}} \in L^2({\hat{\Omega }}_T),{} & {} {\hat{G}} = G{} & {} \hbox { in}\ \Omega _T. \end{aligned} \end{aligned}$$

We can now use Theorem 3.2 to show there exists a unique weak solution $\hat{v}$ to the following problem, on the extended domain

$$\begin{aligned} \begin{aligned}&\partial _{t}\hat{v} =-\gamma \hat{v}+F(\hat{v})+\hat{G}{} & {} \hbox { a.e. in}\ {\hat{\Omega }}_T, \\&\hat{v} = \hat{v}_{0}{} & {} \hbox { a.e. in}\ {\hat{\Omega }} \times \{ t =0 \}, \\ \end{aligned} \end{aligned}$$

such that

$$\begin{aligned} {\hat{v}} = v \quad \text {a.e. in }\Omega _T, \qquad \Vert \hat{v} \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2} +\gamma \Vert \hat{v} \Vert _{L^{2}(\Omega _{T})}^{2} \le C_2, \end{aligned}$$

(6.3)

for some positive constant $C_2$ which depends only on $\Vert \hat{v}_{0}\Vert _{L^{2}(\hat{\Omega })}$, $\Vert \hat{G}\Vert _{L^{2}(\hat{\Omega }_{T})}$ and $K_F$ defined in Lemma 2.2.

We now introduce the following notation: for a function ${\hat{\psi }} :{\hat{\Omega }}_T \rightarrow \mathbb {R}$, and a real number h with $|h| < \delta$, we introduce the function

$$\begin{aligned} {\hat{\psi }}_h :\Omega _T \rightarrow \mathbb {R}, \qquad (x,\xi ,t) \mapsto {\hat{\psi }}(x,\xi +h,t)- {\hat{\psi }}(x,\xi ,t). \end{aligned}$$

The function $\hat{v}_{h}$ satisfies the system

$$\begin{aligned} \begin{aligned}&\partial _{t} \hat{v}_{h} = -\gamma \hat{v}_{h}+ \bigl [ F({\hat{v}}({\hspace{1.111pt}\cdot \hspace{1.111pt}},{\hspace{1.111pt}\cdot \hspace{1.111pt}}+h,{\hspace{1.111pt}\cdot \hspace{1.111pt}}))-F({\hat{v}})\bigr ] +\hat{G}_{h}{} & {} \hbox { in}\ \Omega _T, \\&\hat{v}_{h} =(\hat{v}_{0})_{h}{} & {} \hbox { on}\ \Omega \times \{ t=0 \}. \\ \end{aligned} \end{aligned}$$

Multiplying by $\hat{v}_{h}$, and using Young’s inequality for products, we deduce

$$\begin{aligned} \begin{aligned} \frac{d}{dt}|\hat{v}_{h}|^{2} +\gamma |\hat{v}_{h}|^{2}&= \bigl ([ F(\hat{v}({\hspace{1.111pt}\cdot \hspace{1.111pt}}, {\hspace{1.111pt}\cdot \hspace{1.111pt}}+h, {\hspace{1.111pt}\cdot \hspace{1.111pt}}))-F(\hat{v})] +\hat{G}_{h}\bigr ) \hat{v}_{h} \\&\le \frac{2}{\gamma } \!\Bigl ( | F(\hat{v}({\hspace{1.111pt}\cdot \hspace{1.111pt}}, {\hspace{1.111pt}\cdot \hspace{1.111pt}}+h, {\hspace{1.111pt}\cdot \hspace{1.111pt}}))-F(\hat{v}) |^{2} + |\hat{G}_{h}|^{2} \Bigr ) +\frac{\gamma }{2}|\hat{v}_{h}|^{2} \quad \hbox { a.e. in}\ \Omega _{T} \end{aligned} \end{aligned}$$

Integrating the last deduced inequality over $\Omega \times (0,t)$ using Lemma 2.2 we obtain

$$\begin{aligned} \Vert \hat{v}_{h}(t)\Vert _{L^{2}(\Omega )}^{2} \!\le \! C_3\int _{0}^{t} \left( \!\Vert \hat{v}_{h}(s)\Vert _{L^{2}(\Omega )}^{2} \!+\! \Vert \hat{G}_{h}(s)\Vert _{L^{2}(\Omega )}^{2}\right) ds \!+ \!\Vert (\hat{v}_{0})_{h}\Vert _{L^{2}(\Omega )}^{2}, \, t\in (0,T), \end{aligned}$$

where $C_3$ depends on $\gamma$ and $K_F$. A further application of Gronwall’s inequality gives

$$\begin{aligned} \Vert \hat{v}_{h} \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2} \le e^{C_3 T} C_4 \big ( \Vert ( \hat{v}_{0})_{h} \Vert _{L^{2}(\Omega )}^{2} +\Vert \hat{G}_{h}\Vert _{L^{2}(\Omega _{T})}^{2} \big ), \end{aligned}$$

where the constant $C_4$ depends on T, $\gamma$, and $K_F$.

Applying (Mikhailov 1978, Theorem 4, page 120) and the first identity of (6.3), we conclude that

$$\begin{aligned} \partial _{\xi }v\in L^{\infty }(0,T;L^{2}(\Omega )), \end{aligned}$$

such that

$$\begin{aligned} \Vert \partial _{\xi } v \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2} \le e^{C_3T} C_4 \big ( \Vert \partial _{\xi } v_{0} \Vert _{L^{2}(\Omega )}^{2} +\Vert \partial _{\xi } G \Vert _{L^{2}(\Omega _{T})}^{2}), \end{aligned}$$

(6.4)

with $C_4$ dependent on T, $\gamma$, and $K_F$.

Now that we have bounded $\Vert \partial _\xi v(t) \Vert$ uniformly in t, we return to (6.2). Applying Gronwall’s lemma to the inequality obtained from (6.2) upon disregarding the second term on the left-hand side, and using (6.4), we obtain

$$\begin{aligned} \begin{aligned} \Vert w(t) \Vert ^2_{L^2(\Omega )}&\le \nu \frac{e^{C_1 T}}{2} \Vert \partial _{\xi } v \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}^{2} \\&\le \nu \frac{e^{C_1 T} e^{C_3T}}{2} C_4 \big ( \Vert \partial _{\xi } v_{0} \Vert _{L^{2}(\Omega )}^{2} +\Vert \partial _{\xi } G \Vert _{L^{2}(\Omega _{T})}^{2}), \end{aligned} \end{aligned}$$

which proves the estimate (6.1). $\square$

7 Numerical Results

In this section, we present numerical results in support of the theory presented in the past sections. We simulate system (1.2) on a rescaled domain

$$\begin{aligned} \mathbb {T}= \mathbb {R}/2L_{x}\mathbb {Z}, \qquad \Omega = \mathbb {T}\times (-L_\xi ,L_\xi ), \qquad \Omega _T = \Omega \times [0,T], \end{aligned}$$

that is

$$\begin{aligned} \begin{aligned}&\begin{aligned} \partial _t v(x,\xi ,t) = (-\gamma + \nu \partial _{\xi }^{2})v(x,\xi ,t)&+ G(x,\xi ,t) \\&+F(v({\hspace{1.111pt}\cdot \hspace{1.111pt}},{\hspace{1.111pt}\cdot \hspace{1.111pt}},t))(x,\xi ), \end{aligned}{} & {} (x,\xi ,t) \in \Omega _T, \\&\partial _\xi v(x,-L_\xi ,t) = 0, \qquad \partial _\xi v(x,L_\xi ,t) = 0,{} & {} (x,t) \in \mathbb {T}\times [0,T],\\&v(x,\xi ,0) = v_0(x,\xi ){} & {} (x,\xi ) \in \Omega , \end{aligned} \end{aligned}$$

(7.1)

with the functional and parameter choices similar to Avitabile et al. (2020). In particular, we define

$$\begin{aligned} \delta _\sigma (\xi )= \frac{1}{\sigma \sqrt{\pi }} \exp \biggl ( -\frac{\xi }{\sigma }^2\biggr ), \qquad \sigma \in \mathbb {R}_{>0}, \end{aligned}$$

(7.2)

set the nonlinear integral term F as in (1.3) with

$$\begin{aligned} \begin{aligned}&W(x,\xi ,x',\xi ') = \frac{\kappa }{2}\exp (-|x-x'|)\delta _\sigma (\xi -\xi _0)\delta _\sigma (\xi '),{} & {} \xi _0,\kappa \in \mathbb {R}\\&S(u,\mu ,\theta ) = \frac{1}{1+\exp (-\mu (u-\theta ))},{} & {} \mu \in \mathbb {R}_{ \ge 0}, \theta \in \mathbb {R}, \end{aligned} \end{aligned}$$

(7.3)

and consider null external input, $G \equiv 0$, and a localised initial condition given by

$$\begin{aligned} v_0(x,\xi ) = {\left\{ \begin{array}{ll} \alpha (x) \delta _\sigma (\xi ) &{} \text {if }x >0, \\ \alpha (-x) \delta _\sigma (\xi ) &{} \text {if }x \le 0, \\ \end{array}\right. } \quad \alpha (x) = 1 -S(x,\rho ,x_0), \quad \rho ,x_0 \in \mathbb {R}. \end{aligned}$$

(7.4)

We simulate the system above using the implicit–explicit time stepper derived in Avitabile et al. (2020), with time step size $\tau = 0.05$, and number of spatial grid points $n_x =2^{12}$, $n_\xi = 2^{10}$. The initial condition $v_0$ is spatially localised around the origin. It is close to zero everywhere, except in a small strip, with characteristic length scale $x_0$ and $\sigma$ in the x and $\xi$ direction, respectively. Further, we remark that the synaptic kernel is chosen so that connections are maximal at $\xi = \xi _0$.

We first consider the system in the absence of diffusion, $\nu =0$, as shown in Fig. 2a. At $t=1$, the activity is localised with two peaks, the first one centred at $\xi =0$, and the other at $\xi =\xi _0$: the former is induced by the initial condition $v_0$, while the latter is forming spontaneously, owing to nonlinear effects and driven by the kernel. The dynamics is dissipative for the first bump, while the second peak dominates and persists over long time scales. On the other hand when $\nu$ is small (see Fig. 2b–c), the profiles are merged by diffusion, and the final profile is wider in the $\xi$ direction, as expected.

Fig. 2

Numerical simulation of model (7.1)-(7.4) in a the diffusion-less case $\nu =0$, and b under weak diffusion $\nu =0.1$, with solution profiles for $x=0$ (dashed lines) plotted in (c). In the initial stages of the model without diffusion, the voltage displays a profile with two bumps, which then evolves towards a single bump, centred at $\xi =\xi _0$. As expected, when diffusion is small ($\nu = 0.1$) the two bumps are merged by diffusion, and the final profile is wider and with lower amplitude. Other parameters: $\gamma =0.5$, $\sigma =0.5$, $\kappa =1$, $\xi _0=1$, $\mu =10^3$, $\theta =0.1$, $\rho =5$, $x_0=20$, $L_\xi =3$, $L_x=24\pi$, and $T=3$

Further, we can verify the statement of Theorem 6.1. If we denote by v and $v_\nu$ the solution to the problem with $\nu =0$ and $\nu \ne 0$, respectively, the theorem predicts that the square distance $e(\nu ) = \Vert v - v_\nu \Vert ^2_{L^\infty (0,T;L^2(\Omega ))}$, is an $O(\nu )$ as $\nu \rightarrow 0$. In Fig. 3, we provide numerical evidence of this prediction, by repeating the simulation above for various values of $\{\nu _k\} \subset [0,0.1]$, computing $\{ e(\nu _k) \}$, and fitting the points $\{ (\nu _k,e(\nu _k)) \}$ with a linear regression.

Fig. 3

Square distance between solutions with and without diffusion $e(\nu ) = \Vert v - v_\nu \Vert ^2$, where the norm is on ${L^\infty (0,T;L^2(\Omega ))}$, for various values of $\nu \in [0,0.1]$. Points $(\nu _k, e(\nu _k))$ are fit with a linear regression, providing numerical evidence that $e(\nu ) = O(\nu )$ as $\nu \rightarrow 0$, as predicted by Theorem 6.1. Parameters as in Fig. 2

8 Conclusions

In this paper, we studied the well-posedness and regularity of solutions to neural field models with dendrites. These models are strongly anisotropic, as diffusion acts only on one of the spatial directions (the dendritic direction). The particular laminar structure of the model allows one to characterise solutions as perturbations to a classical (diffusion-less) neural field models. This structure has already been exploited numerically in Avitabile et al. (2020), but was not supported by analytical investigations.

Tackling this problem required studying perturbed weak solutions to the problem, and this opens up the possibility of devising schemes in full generality. The heuristic schemes presented in Avitabile et al. (2020) relied on collocation on a simple domain. The weak formulations studied here are the starting point for the formulation of finite-element schemes, in which the domain $\Omega$ is arbitrary, an avenue that will be explored in future work.

Furthermore, our study shows that solutions to the dendritic neural fields can be approximated, in the limit of small diffusion, with classical neural fields, for which finite-element and spectral methods have recently been developed. We have also provided numerical evidence that this perturbation results hold on O(1) time horizons, in spite of the exponentially growing constants predicted by Theorem 6.1.

This suggests that it should be possible to study splitting schemes in which, at each time interval, a predictor step is carried out using standard neural field models, while corrections are made to solve the original problem with diffusion. The latter problem is relatively cheap to carry out, in view of the anisotropic diffusion operator, which acts solely along $\xi$. It will be of particular interest to try this splitting strategy on problems for which diffusion is not necessarily small, and study the corresponding accuracy of the schemes.