Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK
© Wang et al.; licensee BioMed Central Ltd. 2014
Received: 19 February 2014
Accepted: 7 April 2014
Published: 11 April 2014
Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system’s set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This “code-based” approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution.
As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations.
The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts.
Natural systems exhibit an amazing diversity of patterned structures in both living and nonliving systems, such as animal coats (e.g., zebra stripes, leopard spots and fish spirals), chemicals in a gel, laser light in a cavity, charges on the surface of a semiconductor, ecological balance between two species and neuronal activations in human cortex. Generally, there exist two kinds of natural patterns:
Thermodynamic equilibrium-based, such as crystal structures in inorganic chemistry and spontaneously emergent organic polymer patterns. The forming mechanisms of these patterns have been extensively studied and well-explained via thermodynamics and statistical physics: When a system’s temperature decreases, its entropy and the Gibbs free energy accordingly become smaller (systems tend to move towards a thermodynamic equilibrium where the Gibbs free energy reaches a minimum). Thus the principle of minimum energy allows the system to form certain spatial structures.
Thermodynamics far-from-equilibrium, such as examples mentioned at the beginning of this section. These patterns emerge away from thermodynamic equilibrium, thus thermodynamic theory is not applicable. The understanding of these patterns may require application of kinetic theories.
Pattern dynamics research focuses on universal pattern-forming mechanisms for the latter case. Away from thermodynamic equilibrium, some experimentally observed 2D spatial patterns have been reported: Rayleigh-Bénard convection patterns; reaction-diffusion Turing patterns; Faraday-wave patterns; vibratory granular patterns; slime-mold spiral patterns and Ca2+ spiral waves in Xenopus laevis eggs. Although these patterns are different in the spatiotemporal scales or detailed pattern-forming mechanisms, they have similar structures.
More than half a century ago, British mathematician Alan Turing strove to understand the spontaneous processes generating these structures. In his famous 1952 paper, he proposed a reaction-diffusion model explaining potential mechanism for animal coats: At a certain stage of embryonic development, the reaction and diffusion between molecules, known as morphogens, and other reactors, lead to the breaking of symmetry of the homogeneous state. The morphogens spontaneously evolve to a non-uniform state, leading to the unique textures seen on animal skin. The key pattern-forming idea in Turing’s paper is the system’s spontaneous symmetry-breaking mechanism, also known as a Turing bifurcation (spatial instability).
However, for nearly 30 years Turing’s paper drew little attention for two reasons: morphogens had not been found in biological systems; negative chemical concentrations are permitted in Turing’s model, but are not accepted by chemists. From the late 1960s, a Brussels team led by Ilya Prigogine (winner of 1977 Nobel prize in chemistry) was endeavouring to prove Turing’s theory. Prigogine developed a reaction-diffusion model, Brusselator, to show the existence of Turing patterns that obey the rules of chemical kinetics. The Brusselator is a mathematical representation of the interaction between two morphogens, a reactor and an inhibitor, competitively reacting in time and spreading in space, which could give rise to a symmetry-breaking transition bifurcating from a homogeneous to patterned state, either stationary in a spatial (Turing) structure or in a temporal (Hopf) oscillation. The Brusselator model further revealed the “secret” of Turing patterns: a coupling between nonlinear reaction kinetics and distinct diffusion rates such that the inhibitor should diffuse more rapidly than the activator.
This Brusselator proposed activator–inhibitor interaction is now a well known universal principle explaining regular pattern formation in chemical[17–19], ecological[5, 20] and physical[21, 22] systems, as well as biological systems[23, 24].
As pattern-forming systems essentially consist of coupled differential equations, the simulated patterns are time- and space-dependent solutions of the differential equations. The MATLAB built-in fourth-order Runge-Kutta solver ode45 and custom Euler methods are common ways to integrate differential systems. The implementation of these algorithms are well explained and demonstrated in many studies[25–28]. For example, in Yang’s book, at the end of Part II Yang presents a piece of concise MATLAB code for efficiently simulating simple reaction-diffusion systems. With some modifications, Yang’s programs can be used to simulate pattern formation in a wide range of applications of nonlinear reaction-diffusion equations. Some of these examples are discussed in detail in Part III of Yang’s book. Alternative to MATLAB, there are other options for pattern simulations such as MEREDYS, implemented in the Java programming language, interpreting the NEUROML model description language. Besides, there are other programming environments applicable to modelling pattern formations such as COMSOL and MODELICA.
Although it is efficient to solve differential equations in MATLAB or other programming platforms, their code script-based pattern simulations require high-level programming skills to tune the model parameters in order to examine the pattern dynamics; this limits their uptake by non-programmers. A few attempts for graphic-based pattern simulations have been made in the last decades. For example, READY is an ideal program for getting started with reaction-diffusion systems. It runs fast (taking advantage of multi-core architectures), is easy to use (graphic-based) and runs on multiple platforms. In addition, there are other Java applets allowing easy pattern simulations such as Gray-Scott Simulator, showing the Gray-Scott pattern: phenomena in grids of points that are connected “amorphously”. This closely models the development of a biological system of living cells. Similarly, Lidbeck has created another Java application with extensive presets and options, and even allows 3-D of pattern simulations. However, these graphical user interface (GUI)-based softwares designed mainly for demonstrations of specific (pre-loaded) models. READY supports custom models but is written in READY-specific codes. So technically, READY is still a coding-based platform with a GUI interface. In the community of pattern dynamicists, there may be a demand for a software platform circumventing the programming burden required to implement numerical simulations of biologically-relevant pattern-forming systems.
The aim of this paper is to introduce SIMULINK modelling for simulating pattern dynamics. SIMULINK, an add-on product to MATLAB, provides an interactive, graphical environment for simulating and analysing dynamic systems. It enables modelling via a graphical programming language based on block diagrams. SIMULINK has been used extensively in engineering field such as control theory and digital signal processing for multidomain simulation as well as model-based design. Besides, SIMULINK users have extended its applications in other areas such as medical device prototyping, heat transfer modelling[41, 42] and chemical reactions.
The present paper introduces the application of SIMULINK to pattern simulation studies, and it is organised as follows. We start, for pedagogical reasons, with a brief demonstration of the SIMULINK in modelling differential equations. Then, we construct the well-known Brusselator model in SIMULINK. By extending the Brusselator modelling strategies, we construct a SIMULINK-based human cortical model (developed by the authors’ team) that incorporates the pattern-forming essentials while retaining neurophysiological accuracy: the cortical model comprises interacting populations of excitatory and inhibitory neurons which communicate via chemical (neurotransmitter-controlled) and electrical (gap-junction) synapses. In the model, the interaction between chemical kinetics and electrical diffusion allow for the emergence of a comprehensive range of patterns, which may be of direct relevance to clinically observed brain dynamics such as epileptic seizure EEG spikes, deep-sleep slow-wave oscillations and cognitive gamma-waves[46, 47].
Finally, we examine the Brusselator and cortical model pattern dynamics. These simulation results are compared with the theoretical predictions.
The diffusion constant D U,V [with units (length)2/time] is an important parameter indicative of the diffusion mobility. For a multi-component system, the higher the diffusivity, the faster the species diffuse into each other. Here, f U,V (U,V) is typically a nonlinear function of concentrations U and V.
Solving a pattern-forming system in the form of Eq. (1) requires interpreting the differential operators for time ∂/∂ t and space ∇2. In the following example, we first model the van der Pol oscillator in SIMULINK to explain how we interpret the differential operator on time.
Van der Pol oscillator
We would now integrate these equations with time using the MATLAB numerical integrator ode45. This helps to form the link with the integration in SIMULINK.
At a first glance, the interface for SIMULINK is completely different from the code sheet. In SIMULINK, all calculating elements are displayed by blocks. We select blocks from the SIMULINK library, then connect them to build a model.
The product block (labelled with ×) in Figure2 combines (1 - x 2) and dx/dt. The result is amplified by a gain (triangle block, valued μ), then passed through a function block where x is subtracted. Here, the RHS of Eq. (5) is constructed.
Modelling a differential equation in SIMULINK requires forming a closed loop, where the integrated variables are fed back into the system. Evolution proceeds until reaching the desired final time. The scope block shows the real-time output of the two integrators; the scope can be placed anywhere to monitor the response of a sub-system. The Out1 and Out2 terminals send outputs of two integrators to the MATLAB workspace for further analysis. The results of this SIMULINK model are exactly the same as shown in Figure1.
Readers should be aware of the choice of an appropriate differential solver for a specific problem, depending on the stiffness of differential equations. Applying a wrong solver may lead to either unstable solution or exceptional computation time. However, it is practically difficult to identify the stiffness of a differential model, thus one should try at least two different solvers, and compare the results. If they concur, i.e. give the same solution, they are likely to be correct. As suggested by MATLAB help file, it is worthwhile to try ode45 first since it is the most widely used method. For pattern-forming systems, we can also compare the numerical solution with the theoretical prediction to identify the applicability of the solver. For the demonstrated Brusselator and cortical models, ode45 and ode23 both work well and give rise to similar result; moreover, the numerical solutions match well with the theoretical predictions in emergent patterns (see Results and discussion section). So we choose ode45 solver to integrate the differential-equation models in this paper.
The Brusselator model describes the competition of two chemical species in a chemical reaction, and is the simplest reaction-diffusion system capable of generating complex spatial patterns. The competition between two reactors and the introduction of diffusion satisfy the key requirements for pattern formation. The pattern dynamics of the Brusselator has been comprehensively examined in de Wit and other researchers’ work[51–54]. Here, our purpose is to introduce the SIMULINK modelling strategies.
SIMULINK version of Brusselator model
where X and Y denote concentrations of activator and inhibitor respectively; D X and D Y are diffusion constants; A is a constant and B is a parameter that can be varied to result in a range of different patterns.
In SIMULINK, we initialise the Brusselator model as a column vector consisting of a 60 × 1 grid (spatial resolution = 1 cm/grid-point) for the one-dimensional case; or as a 60 × 60 grid for the two-dimensional case. Grid edges are joined to give toroidal boundaries.
where we have again assumed a square grid so that h x = h y = h.
The reason we introduce the custom block is that the SIMULINK built-in 2-D CONV block provides only the “valid” (non-flux) boundary condition, and cannot handle periodic boundaries.
SIMULINK versions of Waikato cortical model equations
The authors’ team at the University of Waikato (New Zealand) has developed a mean-field cortical model which encapsulates the essential neurophysiology of the cortex, while remaining computationally cost efficient. The model envisions the cortex as a continuum in which pools of neurons are linked via chemical and electrical (gap-junction) synapses.
The Waikato cortical model has a rich history of development: The model is based on early work by Liley et al., with enhancements motivated by Rennie et al. and Robinson et al.; and has been recently extended to include electrical synapses (also referred as gap junctions)[44, 47] supplementing neural communications via chemical synapses.
Cortical element is the “macrocolumn” containing ∼100,000 neurons.
There are only two distinct kinds of cortical neurons: 85% excitatory, and 15% inhibitory neurons. “Excitatory” and “inhibitory” are classified according to their effects on other neurons.
There are three isotropic neuronal interactions: cortico-cortical (long-range from other macrocolumns), intra-cortical (short-range within macrocolumn) and connections from subcortical structures such as the thalamus and brain-stem. A macrocolumn with diameter ∼1 mm and depth ∼3 mm is sketched in Figure 6. We further assume that inhibitory connections via their short axons are locally restricted within a macrocolumn. In contrast, the axons from excitatory neurons are often longer and extensively branched, having length ranging from centimetres to several meters (e.g., in a giraffe’s neck), allowing exclusively excitatory cortico-cortical connections. Both excitatory and inhibitory connections are permitted in local neuron connections.
Neurons exchange information via both chemical and electrical (gap-junction) synapses. Gap-junctions are more abundant within inhibitory populations, while being rare within excitatory neuronal populations.
The authors’ team first introduced gap-junctions into the cortical model in the paper Gap-junction mediate large-scale Turing structures in a mean-field cortex driven by subcortical noise. The cortical model is expected to exhibit similar dynamics to a chemical reaction-diffusion system: The interaction between the excitatory and inhibitory neurons is analogous to to the competition between the activator and inhibitor of the chemical reaction-diffusion model, e.g. the Brusselator. The gap-junction strength between inhibitory neurons also plays the same role as the diffusion terms in the Brusselator, which allows a spatial evolution of the patterns. Consequently, it is reasonable that the cortical model exhibits similar pattern dynamics for those seen in the chemical reaction-diffusion system.
where ξ eb is the Gaussian-distributed white-noise sources being delta-correlated in time and space. Inclusion of white-noise stimulation is motivated by physiological evidence that the cortex requires a continuous background “wash” of input noise to function normally.
We write D bb as the diffusive-coupling strength between electrically adjoined e ↔ e, i ↔ i neuron pairs. To simplify the notation, we write (D 1,D 2) ≡ (D ee ,D ii ).
Symbol definitions for the cortical model
Inverse-length scale for e→b axonal connection
Axonal conduction speed
Maximum firing rate
Sigmoid threshold voltage
Standard deviation for threshold
Long-range e → b axonal connectivity
Local e → b, i → b axonal connectivity
e→b tonic flux entering from subcortex
Neuron time constant
Reversal potential at dendrite
Neuron resting potential
Offset to resting potential
Excitatory synaptic gain
Inhibitory synaptic gain
i ↔ i gap-junction diffusive coupling strength
e ↔ e gap-junction diffusive coupling strength
SIMULINK versions of Waikato cortical model equations
Let us first list the mathematical equations for the Waikato cortical model and examine their characteristics.
The cortico-cortical equationcan be arranged by collecting temporal derivatives to the LHS:(22)
The intra-cortical equationshave different RHS, but their LHS are in the same mathematical pattern:(23)We can move the term γ 2Φ to the RHS of the intra-cortical equations, then the LHS of the intra-cortical equations have the expression:(24)
which is similar to the LHS of the cortico-cortical equation.
The soma equationcan be re-arranged as(25)
In following sections, we detail the SIMULINK implementation of the three subsystems (drawn as three blocks in Figure8) of the cortical model.
Results and discussion
Brusselator pattern simulations in 1-D space
The onsets of four main classes of instability
Eigenvalue (σ q = α + iω)
q ≠ 0
σ q = 0
q = 0
α = 0, ω ≠ 0
q ≥ 0
α = 0, ω ≠ 0 at q = 0
q ≠ 0
α ≥ 0, ω > 0
Brusselator pattern simulations in 2-D space
Brusselator simulation in a 2-D space has more varieties than in a 1-D space. These 2-D patterns will have unique spatial Turing structures that we cannot infer from the LSA. To precisely predict the Turing pattern dynamics, we derived the amplitude equations for the Brusselator model at the onset of a Turing instability. The amplitude equation describes a reduced form of a reaction-diffusion system yet still retains its essential dynamical features. By approximating the analytic solution, the amplitude equation allows precise predictions of the pattern dynamics when the system is near a bifurcation point. In simple words, the amplitude equations may offer a guidance for model parameter settings corresponding to specific patterns, e.g., Turing pattern textures.
For the 2-D spatiotemporal simulation of the Brusselator model, SIMULINK was set to 50-s simulation time in auto time-step mode, allowing the pattern-forming system to organise itself sufficiently. Figure18 predicts the stabilities of stripes (red), H0 (blue) and H π (black) modes for the Turing instability of the Brusselator model. From the range of solid curves, we have the summary of parametric space where a specific mode is stable: The stripes mode is stable when μ s- < μ < μ s+; the hexagonal mode is stable (H π and H0) when μ < μ h- or μ > μ h+. H π and H0 interact at μ p where they exchange mode stability, that is, H π will transit to H0 when μ crosses μ p from its LHS to the RHS.
μ = 0.0495 falling into a range where only H π is stable;
μ = 0.1100 falling into a range where stripes and H π modes coexist;
μ = 0.3994 where only the stripes structure is stable;
μ = 0.7000 again falling into a bistable range where stripes and H0 coexist;
μ = 1.4802 where only H0 is stable.
In Figure18, we see good agreement between SIMULINK simulated patterns and theoretical predictions. Clear H π , stripes and H0 structures are observed at (a), (c) and (e) cases respectively. (b) and (d) show mixed states between forward and backward stable structures. Consequently, we can conclude that by increasing μ, the distance to the critical point, positively, Brusselator forms sequentially from H π , to stripes, to H0.
In summary, to construct the Brusselator model in SIMULINK, we first place an integrator block to represent time derivative. The temporal integrator’s output will be fed back into the system to engage with the system’s evolution, then form the input of this integrator block, closing the loop. The model parameters can be adjusted by tuning the settings of the blocks A and B as well as two gains (labelled D X and D Y respectively). The real-time spatiotemporal evolution of X and Y are monitored via the Matrix viewer block. The simout block delivers the solution of Eq. (6) to the MATLAB workspace for future analysis. The solution is a three-dimensional matrix with the third dimension the same length as the time span.
In the next section, we will implement the SIMULINK-based cortical model and examine its clinically-relevant pattern dynamics.
Cortical model stability and simulations
At the vicinity of a Turing instability, a weak Hopf instability can be induced in parallel by prolonging the timing of delivery of inhibition at chemical synapses. This permits slow Hopf oscillations with the spatial structure maintained. Specifically, a reduction of the inhibitory rate-constant γ i in Eq. (20b) below a critical value ∼30.94 s-1 is sufficient to produce a complex dominant eigenvalue at zero wavenumber whose real part is positive; thus suggesting a global Hopf oscillation.
Steyn-Ross et al. posit that the interacting low-frequency Hopf and Turing instabilities may form the substrate for the cognitive state, namely, the “default” background state for the non-cognitive brain. These slow patterned oscillations may relate to very slow (≤0.1 Hz) fluctuations in BOLD (blood-oxygen-level dependent) signals detected using fMRI (functional magnetic resonance imaging) of relaxed, non-tasked human brains[62, 63]. Steyn-Ross et al. also predict that this default state will be suppressed with elevated levels of subcortical drive during goal-directed tasks[64–68].
The effect of embedding MATLAB functions in SIMULINK running efficiency
Although SIMULINK has an intuitive programming logic and comparable accuracy to MATLAB, it sometimes runs much slower than MATLAB, e.g., in the demonstrated Brusselator and cortical model simulations. The reason is that we embed MATLAB functions wraparound in the model to expand the SIMULINK capability. Once a MATLAB function block is present, the MATLAB interpreter is called at each time-step. This drastically reduces the simulation speed. So, one should use the built-in blocks whenever possible. Without using MATLAB function blocks, SIMULINK shows a higher performance than MATLAB, e.g., see the description of Figure3. MathWorks Support Team also presented comprehensive guidance to speed up the SIMULINK simulation, which are available athttp://www.mathworks.com/matlabcentral/answers/94052. In the further optimisation of our SIMULINK model, we will consider replacing MATLAB function with the MEX S-function, which may help to accelerate the simulation in the merit of its direct communication with the SIMULINK engine (avoid the time consuming compile-link-execute cycle). The build of MEX S-function requires higher level programming skills, so we only address a more accessible method (embedding MATLAB function) for non-experts in this paper.
The strategy for modelling differential equation models in SIMULINK is addressed in this paper. To construct a system of differential equations in SIMULINK, we can directly convert the mathematical terms and operators to the SIMULINK graphical block diagrams. The key idea for programming differential systems in SIMULINK is to form a closed loop, such that the solution of the system can evolve in this loop. The accuracy and reliability of the SIMULINK modelling method has been examined via comparing a van der Pol oscillator represented in MATLAB code-script and SIMULINK block-diagram, showing the SIMULINK model has comparable performance with the code-script version.
Using a well-known Brusselator model, we demonstrated two main SIMULINK modelling strategies for a reaction-diffusion system: interpretation of 2-D convolution with the periodic boundary condition; hybrid programming in SIMULINK with MATLAB functions. The pattern simulations of the Brusselator in SIMULINK are in good agreement with the predictions via bifurcation theories.
Following the pattern-forming theories, we introduced a cortical model with competitive neuronal interactions and diffusions. Unlike the simple Brusselator, the cortical model is a complicated system comprised of distinct cortical connections. Here, we showed how to build these subsystems in SIMULINK. Finally, we connected all subsystems to form the completed Waikato cortical model. The simulations of the cortical model exhibit Turing and mixed Turing–Hopf patterns, the clinical relevance of which is briefly discussed. As the main aim of this paper is to introduce SIMULINK modelling, readers are referred to Steyn-Ross et al.’s recent publications[6, 44–47] for comprehensive investigations of the cortical model.
a vanderpoldemo is a MATLAB pre-coded function
bfixed time-step ODE solvers are not built into MATLAB, but they can be acquired from a release by the MathWorks Support Team:http://www.mathworks.com/matlabcentral/answers/uploaded_files/5693/ODE_Solvers.zip
cThe two-dimensional circular convolution algorithm was written by David Young, Department of Informatics, University of Sussex, UK. His convolve2() code can be downloaded from MathWorks File Exchange:http://www.mathworks.com/matlabcentral/fileexchange/22619-fast-2-d-convolution
d MATLAB simulation codes were written by Alistair Steyn-Ross. The complete codes, plus README files and movies of cortical dynamics, are available from the web site:http://www2.phys.waikato.ac.nz/asr/
We acknowledge supports from the New Zealand–Japan Exchange Programme (NZJEP), and Embedded Systems Lab in Gunma University, Japan.
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