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December  2020, 7(2): 271-290. doi: 10.3934/jcd.2020011

## Numerical investigation of a neural field model including dendritic processing

 1 Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands 2 Mathneuro Team, Inria Sophia Antipolis, 2004 Rue des Lucioles, 06902 Sophia Antipolis, Cedex 3 Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK 4 CEMAT, Instituto Superior Tecnico, University of Lisbon, Portugal

Received  October 2019 Published  July 2020

We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finite-difference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

Citation: Daniele Avitabile, Stephen Coombes, Pedro M. Lima. Numerical investigation of a neural field model including dendritic processing. Journal of Computational Dynamics, 2020, 7 (2) : 271-290. doi: 10.3934/jcd.2020011
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Schematic of the neural field model. (a) Dendrites are represented as unbranched fibres (red), orthogonal to a continuum of somas (somatic layer, in grey). (b) Model (1) is for a 1D somatic layer, with coordinate $x \in \mathbb{R}$, and fiber coordinate $\xi \in \mathbb{R}$. Input currents are generated in a small neighbourhood of the somatic layer, at $\xi = 0$ and are delivered to a contact point, in a small neighbourhood of $\xi = \xi_0$. The strength of interaction depends on the distance between sources and contact points, measured along the somatic layer, hence the inputs that are generated at $A$ and transmitted to $B$, $C$, and $D$ depend on $|x_B - x_A|$, $|x_C-x_A|$, and $|x_D - x_A|$, respectively (see (2))
Coherent structure observed in time simulation of (3), (22), (23). (a): Pseudocolor plot of $V(x,\xi,t)$ at several time points, showing two counter-propagating waves. (b): Solution at $\xi = 0$, showing the wave profile. Parameters: $\xi_0 = 1$, $\varepsilon = 0.005$, $\nu = 0.4$, $\gamma = 1$, $\beta = 1000$, $\theta = 0.01$, $\kappa = 3$, $L_x = 24 \pi$, $L_\xi = 3$, $n_x = 2^{10}$, $n_\xi = 2^{12}$, $\tau = 0.05$
(a) Travelling wave speed versus firing rate threshold, computed analytically via (24), and numerically via the time-stepper. (b)-(d) Convergence of the computed speed to the analytical speed at $\theta = 0.01$, as a function of the the kernel support parameter $\varepsilon$, the steepness of the sigmoid $\beta$, and the time-stepping parameter $\tau$, respectively. Parameters as in (2)
Numerical simulation of Turing bifurcation for the model with kernel and firing rate function given in (28). (a): Plots of $\hat w(p) - w_*$ for $\beta = 28$ and $\beta = 30$, from which we deduce that a Turing bifurcation occurs for an intermediate value of $\beta$. We expect perturbations of the trivial state to decay exponentially if $\beta = 28$ and to increase exponentially if $\beta = 30$, as confirmed in panels (b)-(d). (b): Maximum absolute value of the voltage at $\xi = 0$, as function of time, when the trivial steady state is perturbed. (c), (d): Pseudocolor plot of $V$ when $\beta = 28$ and $\beta = 30$, respectively. Parameters: $\xi_0 = 1$, $\varepsilon = 0.005$, $\nu = 6$, $c = 1$, $a_1 = 1$, $b_1 = 1$, $a_2 = 1/4$, $b_2 = 1/2$, $n_x = 2^9$, $L_x = 10\pi$, $n_\xi = 2^{11}$, $L_\xi = 2.5 \pi$, $\tau = 0.01$
 Algorithm 1: IMEX time stepper in matrix form (8), nonlinear term computed with pseudospectral evaluation (21) Input: Initial condition $V^0 \in \mathbb{R}^{n_\xi \times n_x}$, time step $\tau$, number of steps $n_t$. Output: An approximate solution $(V^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi \times n_x}$ 1 begin 2      Compute grid vectors $\xi \in \mathbb{R}^{n_\xi \times 1}$, $x \in \mathbb{R}^{1 \times n_x}$. 3       Compute synaptic vectors $w, \hat w = \mathcal{F}_{n_x}[w] \in \mathbb{R}^{1 \times n_x}$. 4       Compute synaptic vectors $\alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1}$. 5       Compute quadrature weights $\sigma \in \mathbb{R}^{n_\xi \times 1}$. 6       Compute sparse $LU$-factorisation of $A$, $LU = A \in \mathbb{R}^{n_\xi \times n_\xi}.$ 7       for $n = 1,\ldots,n_t$ do 8              Set $V = V^{n-1} \in \mathbb{R}^{n_\xi \times n_x}$. 9               Compute the external input at time $t_{n-1}$ and store it in $G \in \mathbb{R}^{n_\xi \times n_x}$. 10               Set $z = \mathcal{F}_{n_x}\big[(\alpha' \odot \sigma)^T S(V)\big] \in \mathbb{R}^{1 \times n_x}$. 11                Set $N = h_x \alpha \mathcal{F}^{-1}_{n_x} [ \hat w \odot z] \in \mathbb{R}^{n_\xi \times n_x}$. 12                Solve for $V^{n}$ the linear problem $(LU)V^n = V + \tau(N+G)$. 13      end 14 end
 Algorithm 1: IMEX time stepper in matrix form (8), nonlinear term computed with pseudospectral evaluation (21) Input: Initial condition $V^0 \in \mathbb{R}^{n_\xi \times n_x}$, time step $\tau$, number of steps $n_t$. Output: An approximate solution $(V^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi \times n_x}$ 1 begin 2      Compute grid vectors $\xi \in \mathbb{R}^{n_\xi \times 1}$, $x \in \mathbb{R}^{1 \times n_x}$. 3       Compute synaptic vectors $w, \hat w = \mathcal{F}_{n_x}[w] \in \mathbb{R}^{1 \times n_x}$. 4       Compute synaptic vectors $\alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1}$. 5       Compute quadrature weights $\sigma \in \mathbb{R}^{n_\xi \times 1}$. 6       Compute sparse $LU$-factorisation of $A$, $LU = A \in \mathbb{R}^{n_\xi \times n_\xi}.$ 7       for $n = 1,\ldots,n_t$ do 8              Set $V = V^{n-1} \in \mathbb{R}^{n_\xi \times n_x}$. 9               Compute the external input at time $t_{n-1}$ and store it in $G \in \mathbb{R}^{n_\xi \times n_x}$. 10               Set $z = \mathcal{F}_{n_x}\big[(\alpha' \odot \sigma)^T S(V)\big] \in \mathbb{R}^{1 \times n_x}$. 11                Set $N = h_x \alpha \mathcal{F}^{-1}_{n_x} [ \hat w \odot z] \in \mathbb{R}^{n_\xi \times n_x}$. 12                Solve for $V^{n}$ the linear problem $(LU)V^n = V + \tau(N+G)$. 13      end 14 end
 Algorithm 2: IMEX time stepper in vector form (7), nonlinear term evaluated with quadrature formula (12). Input: Initial condition $U^0 \in \mathbb{R}^{n_\xi n_x}$, time step $\tau$, number of steps $n_t$. Output: An approximate solution $(U^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi n_x}$ 1 begin 2      Compute grid vectors $\xi \in \mathbb{R}^{n_\xi}$, $x \in \mathbb{R}^{n_x}$. 3       Compute synaptic vector $w \in \mathbb{R}^{1 \times n_x}$. 4       Compute synaptic vectors $\alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1}$. 5       Compute quadrature weights $\rho \in \mathbb{R}^{n_x}$, $\sigma \in \mathbb{R}^{n_\xi}$. 6      Compute sparse $LU$-factorisation of A, $LU = \big( (1+\tau \gamma) I_{n_x n_\xi} - \tau \nu I_{n_x} \otimes D_{\xi \xi} \big) \in \mathbb{R}^{n_\xi n_x \times n_\xi n_x }.$ 7      for $n = 1,\ldots,n_t$ do 8              Set $Z = U^{n-1} \in \mathbb{R}^{n_\xi n_x}$. 9              Compute the external input at time $t_{n-1}$ and store it in $G \in \mathbb{R}^{n_\xi n_x}$. 10              Compute the nonlinear term $N$ using (12). 11              Solve for $U^{n}$ the linear problem $(LU)U^n = Z + \tau(N+G)$. 12      end 13 end
 Algorithm 2: IMEX time stepper in vector form (7), nonlinear term evaluated with quadrature formula (12). Input: Initial condition $U^0 \in \mathbb{R}^{n_\xi n_x}$, time step $\tau$, number of steps $n_t$. Output: An approximate solution $(U^n)_{n=1}^{n_t} \subset \mathbb{R}^{n_\xi n_x}$ 1 begin 2      Compute grid vectors $\xi \in \mathbb{R}^{n_\xi}$, $x \in \mathbb{R}^{n_x}$. 3       Compute synaptic vector $w \in \mathbb{R}^{1 \times n_x}$. 4       Compute synaptic vectors $\alpha, \alpha' \in \mathbb{R}^{n_\xi \times 1}$. 5       Compute quadrature weights $\rho \in \mathbb{R}^{n_x}$, $\sigma \in \mathbb{R}^{n_\xi}$. 6      Compute sparse $LU$-factorisation of A, $LU = \big( (1+\tau \gamma) I_{n_x n_\xi} - \tau \nu I_{n_x} \otimes D_{\xi \xi} \big) \in \mathbb{R}^{n_\xi n_x \times n_\xi n_x }.$ 7      for $n = 1,\ldots,n_t$ do 8              Set $Z = U^{n-1} \in \mathbb{R}^{n_\xi n_x}$. 9              Compute the external input at time $t_{n-1}$ and store it in $G \in \mathbb{R}^{n_\xi n_x}$. 10              Compute the nonlinear term $N$ using (12). 11              Solve for $U^{n}$ the linear problem $(LU)U^n = Z + \tau(N+G)$. 12      end 13 end
Flop count for the initialisation step (lines 2-6) and for one time step (lines 8-12) in Algorithms 1, 2
 Algorithm 1 Algorithm 2 Lines Flops Lines Flops 2 $n_\xi + n_x$ 2 $n_\xi + n_x$ 3 $2n_x$ 3 $n_x$ 4 $2n_\xi$ 4 $2n_\xi$ 5 $n_\xi$ 5 $n_\xi + n_x$ 6 $2n_\xi-1$ 6 $2n_\xi n_x -1$ 2-6 $O(n_\xi) + O(n_x)$ 2-6 $O(n_\xi n_x)$ 8 $n_\xi n_x$ 8 $n_\xi n_x$ 9 $n_\xi n_x$ 9 $n_\xi n_x$ 10 $3 n_\xi n_x + O(n_x \log n_x) + n_\xi - n_x$ 10 $2n^2_\xi n_x^2 - n^2_\xi n_x$ 11 $n_\xi n_x + O(n_x \log n_x) + 2n_x$ 11 $5n_\xi n_x -4$ 12 $5 n_\xi n_x - 4 n_x$  8-12 $O(n_\xi n_x) + O(n_x \log n_x)$ 8-11 $O(n^2_\xi n^2_x)$
 Algorithm 1 Algorithm 2 Lines Flops Lines Flops 2 $n_\xi + n_x$ 2 $n_\xi + n_x$ 3 $2n_x$ 3 $n_x$ 4 $2n_\xi$ 4 $2n_\xi$ 5 $n_\xi$ 5 $n_\xi + n_x$ 6 $2n_\xi-1$ 6 $2n_\xi n_x -1$ 2-6 $O(n_\xi) + O(n_x)$ 2-6 $O(n_\xi n_x)$ 8 $n_\xi n_x$ 8 $n_\xi n_x$ 9 $n_\xi n_x$ 9 $n_\xi n_x$ 10 $3 n_\xi n_x + O(n_x \log n_x) + n_\xi - n_x$ 10 $2n^2_\xi n_x^2 - n^2_\xi n_x$ 11 $n_\xi n_x + O(n_x \log n_x) + 2n_x$ 11 $5n_\xi n_x -4$ 12 $5 n_\xi n_x - 4 n_x$  8-12 $O(n_\xi n_x) + O(n_x \log n_x)$ 8-11 $O(n^2_\xi n^2_x)$
Space requirements, measured in Floating Point Numbers, for Algorithms 1 and 2. Arrays $d_1, \ldots, d_3$, store diagonals of the $LU$-factorisation in the respective algorithms
 Floating Point Numbers Algorithm 1 Algorithm 2 $n_\xi$ $\xi,\alpha,\alpha',d_1,d_2,d_3,z$ $\xi, \rho, \alpha, \alpha'$ $n_x$ $x,w,\sigma$ $x, \sigma, w$ $n_\xi n_x$ $V,V^n,G,N$ $U^n,Z,N,G,d_1,d_2,d_3$ Total $4n_x n_\xi + 7n_\xi + 3n_x$ $7 n_x n_\xi +2n_\xi + 2n_x$
 Floating Point Numbers Algorithm 1 Algorithm 2 $n_\xi$ $\xi,\alpha,\alpha',d_1,d_2,d_3,z$ $\xi, \rho, \alpha, \alpha'$ $n_x$ $x,w,\sigma$ $x, \sigma, w$ $n_\xi n_x$ $V,V^n,G,N$ $U^n,Z,N,G,d_1,d_2,d_3$ Total $4n_x n_\xi + 7n_\xi + 3n_x$ $7 n_x n_\xi +2n_\xi + 2n_x$
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