Biological Pattern Formation with Cellular Neural Networks

Transcription

1 CNNA 96: Fourth / E lnternational Workshop on Cellular Neural Networks and their Applications, Seville, Spain, June 24-26, 1996 CNN;~ Biological Pattern Formation with Cellular Neural Networks Gianluca Sett;* and Patrick Thirad * Department of Electronics (DEIS) University of Bologna, Viale Risorgimento 2, Bologna, Italy. gsetti@deis.unibo.it Department of Electrical Engineering Swiss Federal Institute of Technology, CH-1015 Lausanne, Switzerland. thiran@circ.de.epfl.ch ABSTRACT: The ability of the conventional, simple Cellular Neural Network (CNN) model to form spatial and spatia-temporal patterns has been recently highlighted [l, 21. This paper continues this investigation, by studying new properties of pattern formation, and their application to modeling of biological pattern forming systems. 1 Introduction In the last few decades a lot of attention has been devoted to the study of pattern formation in the fields of physics, chemistry and, especially biology. Already in 1952, Turing [3] proposed a model to explain the morphogenesis process. Another well-known nonlinear second-order reaction-diffusion model is the Gierer-Meinhardt system [4] 8A t = F(A,B)+DAV2A i?b at = G(A,B)+D~v~B where A and B are the chemicals concentrations of the so-called activator and inhibitor, F(.) and G(.) are nonlinear functions and DA and DB are respective diffusion coefficients. The activator provides the initial instability necessary to create the pattern if the spatial domain has reached a certain size, while the inhibitor eventually supplies stability once the spatial pattern is formed. Both chemicals diffuse spatially, but, in order to stabilitse the pattern, it is necessary that the inhibitor diffuses much faster than the activator (DE >> DA). Lattice dynamical systems, namely large arrays of coupled nonlinear dynamical systems, in which each cell is typically described by a second or third order continuous-time system [6], can also realize activator-inhibitor schemes. Recently, the ability to form both spatial and spatio-temporal patterns has been highlighted in the case of CNNs [l, 21, even when the initial simple Chua & Yang model is used [5]. Our aim, in the present paper, is to continue in this line of investigation, for an autonomous CNN described by with initial condition xi,j(o), time- and space-independent bias I and without any input. The template coefficients Ar,, are space-invariant in this paper, and f(x) = ( I C x - ll)/z. The output of a cell Cij, defined as yi,, = f(xi,,), is thus always bounded: ly,,jl 5 1. A cell Ci,j such that -1 5 xi,, 5 1 will called a linear cell or linear pixel and will be represented by a gray square in the W96/$ IEEE. 279

2 MODELLING OF BIOLOGICAL SYSTEMS ' figures. If it does not operate in the linear zone, it will be called a saturated cell or satuyated pzzel and will be represented by a black square if zi,j > 1 and a white square if xi,j < Pattern formation properties of CNNs A correct spatial discretization of the reaction-diffusion model (1) would results in a lattice of second order systems. We will see however that a number of patteras explained by (1) can be obtained with (2), although the CNN is only made of simpler first order cells. 2.1 One dimensional CNNs A complete characterization of all patterns that can be created by a CNN made a an infinite number of cells is only available up to now for the template [l] A = [6 p SI. (3) Since the A template is symmetric, the CNN is completely stable [5]. Take s > 0. The case Ip- 1 > 2s is not very interesting, as it yields either that the origin is globally asymptotically stable (if p - 1 < -2s), or on the contrary, that any binary sequence of saturated outputs corresponds to a stable equilibrium (if p - 1 > 2s). The case Ip - 11 < 2s leads however to much more interesting structures, as the stable equilibria appear as a succession of strings of at least two adjacent saturated cells, separated by boundaries of B linear cells, where B is the integer satisfying q-2 < B < 1)-1 with q =?r/[?r-arc cos((p-l)/2s)]. With fixed zero-valued Dirichlet boundary conditions and a finite number of cells N, the origin is globally asymptotically stable as long as the length of the array N < q - 1, but unstable when N 2 1, so that a pattern will be created. One of the most useful techniques employed in the study of pattern formation is the mode decompe sition of the linearized system near the origin [4, 71. Defining a convolution mask by p-1 for i=o otherwise the linearized CNN system around the origin can be written as (4) &i(t) = ai * xi(t) (5) where * denotes spatial convolution. Now, if one considers the Discrete Space Fourier Transform (DSFT) of both members of Eq. (4), one gets &(t) = dn%n(t), (6) where Zln is the Discrete Space Fourier Transform (DSFT) of the convolution mask (4), namely When the initial condition is a random perturbation around the origin, the modes, whose gain Bn is positive, will grow in time, contrary to the modes for which the gain is negative. The coupling induces therefore the instability necessary for the unstable spatial modes to grow, whereas the nonlinearity eventually supplies the stability necessary to the pattern formation. The absence of inhibition restricts however the types of pattrns that can be created. Although the considered spatial frequency approach is exact only when all cells are linear, it gives results that can be valid also for the final steady state of the nonlinear system [

3 2.2 Two dimensional CNNs For a 2-dim. CNN of NI x Nz cells described by a general 3 x 3 isotropic template A = [! ;,I, it becomes much more difficult to characterize the complete set of stable patterns. It is however easy to identify three types of motifs that can appear in a stable pattern [l], namely plain, constant patches, if and only if p s + 4r > 0; checkerboard patches, if and only if p - 1-4s + 4r > 0; and vertical and horizontal stripes, if and only if p - 1-4r > 0. The overall pattern will depend on the ability of the motifs to form stable boundaries between them. Again, only patterns with unregular patches of various sizes can be obtained when the nearest-neighbour connections s, r are positive. Although these are found on many animals, the reaction-diffusion model (1) allows a much wider variety of patterns, thanks to a fast diffusing inhibitor. To take this aspect into account, we must use a 5 x 5 template, whose nearest neighbor interactions are positive (local activation) but whose second nearest neighbor interactions are negative (long range inhibition). A rigorous analysis becomes complex to carry out in general cases, but a lot of insight on the pattern formation properties can be obtained by using the spectral approach introduced in the Section 2.1. Defining the convolution mask in the 2-dim. case ai,j as extension of mask definition (4) to the present case, and following similar procedure, one obtains that the system solution can be expressed as Ial,n,(t) = Za,,a2(O)eZn1*nat, where iin,,n, is the 2-dim. DSFT of aij. Similar consideration about the modes stability can be therefore extended to this case. Even in the 2-dim. case, the result of the approximated spectral analysis can be in perfect agreement with the one obtained with a nonlinear analysis; namely, if one consider template (8) with s = 1 and p = r = 0, only the first of the previous inequalities is satisfied, so that patches are the only stable motif that can be formed by the CNN. This perfectly correspond to the spectral analysis result, since in this case the DSFT of (8) shows a lowpass characteristic. The local activation and long range inhibition appearing in our 5 x 5 template make it equivalent to a band-pass filter. Supposing that the coupling associated with a particular template has provided the necessary instability of spatial frequency, three qualitative epochs can be observed in the dynamics of pattern formation [l]: 1) Linear system leading to noise shaping. 2) Local separation of modes into regions of saturated cells. After some time, most cells get saturated, and form regions of one of the three motifs mentioned above. 3) Boundary negoliatzon leading to a globally stable pattern. The last step is, depending on the template, a long process during which the boundaries shapes between the regionally stable motifs are modified. When one unstable mode dominates the others and when the regionally stable patches can form stable boundaries, most of the pattern formation starting from small random initial conditions occurs in the first epoch, and can therefore be well described by this spectral approach. 3 Biological patterns Some of the most spectacular biological patterns that can be well described by a reaction-diffusion mechanism are mammalian coat patterns [I. Thanks to the spatial frequency approach considered in the previous Section, it is possible to design the template parameters in order to isolate a single motif in the pattern that is obtained. For animal coat markings, these are plain patches, stripes and spots. A typical phenomenon encountered in many spotted animals (especially felines) is a transition from spots, on the main body of the animal, to stripes, on its tail. The leopard, the cheetah, the genet show this feature, that is even present in the common cat (see Fig 2). Such a transition can be obtained with a CNN by using the 5 x 5 template [ A= , (9)

4 MODELLING OF BIOLOGICAL SYSTEMS Figure 1: DSFT of the convolution mask corresponding to (a): the A template (9) and (b): to A template (10) whose corresponding DSFT, presented in Fig. 3 (a), has an annulus of unstable modes ensuring a spatial bandpass filter behavior. In fact, by taking a space-varying input bias Ii,j = -(j - 1)(Ni - 1) for an NI x N2 as well, one can obtain a transition between squiggles and spots [2]. However, this is perhaps less biologically plausible, as there is no reason why an external signal woulc affect specific parts of the embryo and not others during the development of its coat marking. The explanation that is usually provided in biology is that the pattern is governed by the geometry of the reaction-diffusion domain 171, in other words by the boundary conditions. The development of spots requires a large domain, whereas a narrow cylinder, that mimic the terminal part of the tail, enforces the creation of stripes. The same behaviour can be obtained by using template (9) (an similar ones) with a bias, if a CNN with the proper array layout and the proper boundary conditions is considered. Fig. 3 shows the result of a simulation computed using a CNN composed by 7 blocks of 20 to 20 x 4 cells, connected togheter in order to approximate a tail shape. Boundary conditions have been chosen as periodic (top and down) and reflective (left and right), and the initial consitions as random with uniform distribution between (-0.1,O.l). One other typical marking coat is the one of the tiger, as reported in Fig. 4. As can be seen stripe patterns are present both in the tail and in the body, but they are characterized by a different thickness and, sometimes, orientation. Moreover in the body part the stripes some sort of stripe bifurcation is also present. The achievement of a similar result with a CNN, has been possible by using the 5 x 5 template A = [ : 0 0 ; :], with the bias I = As can be seen from the DSFT of the correspondent convolution mask, a CNN with template (10) acts, contrary to the previous examples, as a non isotropic filter, namely as a lowpass filter in one direction and as a bandpass in the second one. The results of the simulations of a CNN with 60 cells in the body part and a tail composed by four blocks of 20 x M cells (M = 10,8,6,4) is shown in Fig. 5. Initial conditions were again small random values, while the boundary conditions are periodic (up and down) and serwvalued Dirichlet (left and right). 282

5 Figure 2: A common cat showing a striped tail (and even legs) and a spotted body (left -from a picture by Yann Artbus - Bertrand), and a typical leopard tail (middle: pre-natal, right: adult - from [7]). Figure 3: Pattern generated from random initial conditions with uniform distribution between (-0.1,O.l) by using space-invariant template (9) and a bias I = The transition from spots to stripes is only dictated by the boundary conditions, that are periodic (up and down) and reflective (left and right). One-dimensional templates can also produce similar patterns as the one encountered on some seashells, where the temporal development of the pattern is somehow recorded as the shell growth and the pattern formation occur simultaneously. Symmetric templates, with positive nearest neighbor and negative second nearest neighbor interactions, can create periodic patterns in space, which result in lines parallel to the direction of growth of the shell (see Chapter 2 of [SI), while anti-symmetric templates can create travelling waves [2], which account for oblique lines with respect to direction of growth (see Chapter 3 of [SI). Much more complex patterns, that sometimes need one activator and two inhibitors instead of one, can however be found (travelling waves in two opposite directions, annihilating each other) and may be too complex to create with the simple model (2). It is of course an open question to determine what pattern complexity this model can create. 283

6 MODELLING OF BIOLOGICAL SYSTEMS Figure 4: Typical coat marking of a tiger: note how stripes are present both in the tail and in the body. - from [7]. Figure 5: Pattern generated from random initial conditions with uniform distribution between (-0.1,O.l) by using the space-invariant template (10) and bias I = The boundary conditions are periodic (up and down) and Dirichlet with zero value (left and right). References [l] P. Thiran, K. R. Crounse, L. 0. Chua and M. Hasler: Pattern formation properties of Autonomous Cellular Neural Networks, IEEE Bansactions on Circuits and Systems I, CAS-42, pp , [2] [3] [4] [5] [6] [7] [8] K. R. Crounse, P. Thiran, G. Setti and L. 0. Chua: Characterization and Dynamics of Pattern Formation in Cellular Neural Networks, Int. J. Bzf. Chaos, to appear, A. M. Turing: The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, 237(B), pp , H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, L.O. Chua and L. Yang, Cellular Neural Networks: Theory, IEEE Pansactions on Circuits and Systems, vol. CAS-35, pp ,1988. L. Goras and L. 0. Chua: Turing Patterns in CNNs - Part 11: Equation and Behaviors, ZEEE Trans. Circuits Syst-I, CAS-42, pp , Oct J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, H. Meinhardt, The Algorathmic Beauty of Sea Shells, Springer-Verlag, Berlin,