Bayesian parameter identification for Turing systems on stationary and evolving domains

In this study, we apply the Bayesian paradigm for parameter identification to a well-studied semi-linear reaction-difusion system with activatordepleted reaction kinetics, posed on stationary as well as evolving domains. We provide a mathematically rigorous framework to study the inverse problem of finding the parameters of a reaction-diffusion system given a final spatial pattern. On the stationary domain the parameters are finite dimensional, but on the evolving domain we consider the problem of identifying the evolution of the domain, i.e. a time dependent function. While others have considered these inverse problems using optimisation techniques, the Bayesian approach provides a rigorous mathematical framework for incorporating the prior knowledge on uncertainty in the observation and in the parameters themselves, resulting in an approximation of the full probability distribution for the parameters, given the data. Furthermore, using previously established results, we can prove wellposedness results for the inverse problem, using the well-posedness of the forward problem. Although the numerical approximation of the full probability is computationally expensive, parallelised algorithms make the problem solvable using high-performance computing

A Bayesian Approach to Modelling Biological Pattern Formation with Limited Data

Pattern formation in biological tissues plays an important role in the development of living organisms. Since the classical work of Alan Turing, a pre-eminent way of modelling has been through reaction-diffusion mechanisms. More recently, alternative models have been proposed, that link dynamics of diffusing molecular signals with tissue mechanics. In order to distinguish among different models, they should be compared to experimental observations. However, in many experimental situations only the limiting, stationary regime of the pattern formation process is observable, without knowledge of the transient behaviour or the initial state. The unstable nature of the underlying dynamics in all alternative models seriously complicates model and parameter identification, since small changes in the initial condition lead to distinct stationary patterns. To overcome this problem the initial state of the model can be randomised. In the latter case, fixed values of the model parameters correspond to a family of patterns rather than a fixed stationary solution, and standard approaches to compare pattern data directly with model outputs, e.g., in the least squares sense, are not suitable. Instead, statistical characteristics of the patterns should be compared, which is difficult given the typically limited amount of available data in practical applications. To deal with this problem, we extend a recently developed statistical approach for parameter identification using pattern data, the so-called Correlation Integral Likelihood (CIL) method. We suggest modifications that allow increasing the accuracy of the identification process without resizing the data set. The proposed approach is tested using different classes of pattern formation models. For all considered equations, parallel GPU-based implementations of the numerical solvers with efficient time stepping schemes are provided.Comment: More compact version of the text and figures, results unchange

Why Adversarial Interaction Creates Non-Homogeneous Patterns: A Pseudo-Reaction-Diffusion Model for Turing Instability

Long after Turing's seminal Reaction-Diffusion (RD) model, the elegance of his fundamental equations alleviated much of the skepticism surrounding pattern formation. Though Turing model is a simplification and an idealization, it is one of the best-known theoretical models to explain patterns as a reminiscent of those observed in nature. Over the years, concerted efforts have been made to align theoretical models to explain patterns in real systems. The apparent difficulty in identifying the specific dynamics of the RD system makes the problem particularly challenging. Interestingly, we observe Turing-like patterns in a system of neurons with adversarial interaction. In this study, we establish the involvement of Turing instability to create such patterns. By theoretical and empirical studies, we present a pseudo-reaction-diffusion model to explain the mechanism that may underlie these phenomena. While supervised learning attains homogeneous equilibrium, this paper suggests that the introduction of an adversary helps break this homogeneity to create non-homogeneous patterns at equilibrium. Further, we prove that randomly initialized gradient descent with over-parameterization can converge exponentially fast to an $\epsilon$-stationary point even under adversarial interaction. In addition, different from sole supervision, we show that the solutions obtained under adversarial interaction are not limited to a tiny subspace around initialization.Comment: 35th AAAI Conference on Artificial Intelligenc

Subbarrel patterns in somatosensory cortical barrels can emerge from local dynamic instabilities

Complex spatial patterning, common in the brain as well as in other biological systems, can emerge as a result of dynamic interactions that occur locally within developing structures. In the rodent somatosensory cortex, groups of neurons called "barrels" correspond to individual whiskers on the contralateral face. Barrels themselves often contain subbarrels organized into one of a few characteristic patterns. Here we demonstrate that similar patterns can be simulated by means of local growth-promoting and growth-retarding interactions within the circular domains of single barrels. The model correctly predicts that larger barrels contain more spatially complex subbarrel patterns, suggesting that the development of barrels and of the patterns within them may be understood in terms of some relatively simple dynamic processes. We also simulate the full nonlinear equations to demonstrate the predictive value of our linear analysis. Finally, we show that the pattern formation is robust with respect to the geometry of the barrel by simulating patterns on a realistically shaped barrel domain. This work shows how simple pattern forming mechanisms can explain neural wiring both qualitatively and quantitatively even in complex and irregular domains. © 2009 Ermentrout et al

Isolating and Quantifying the Role of Developmental Noise in Generating Phenotypic Variation

Genotypic variation, environmental variation, and their interaction may produce variation in the developmental process and cause phenotypic differences among individuals. Developmental noise, which arises during development from stochasticity in cellular and molecular processes when genotype and environment are fixed, also contributes to phenotypic variation. While evolutionary biology has long focused on teasing apart the relative contribution of genes and environment to phenotypic variation, our understanding of the role of developmental noise has lagged due to technical difficulties in directly measuring the contribution of developmental noise. The influence of developmental noise is likely underestimated in studies of phenotypic variation due to intrinsic mechanisms within organisms that stabilize phenotypes and decrease variation. Since we are just beginning to appreciate the extent to which phenotypic variation due to stochasticity is potentially adaptive, the contribution of developmental noise to phenotypic variation must be separated and measured to fully understand its role in evolution. Here, we show that variation in the component of the developmental process corresponding to environmental and genetic factors (here treated together as a unit called the LALI-type) versus the contribution of developmental noise, can be distinguished for leopard gecko (Eublepharis macularius) head color patterns using mathematical simulations that model the role of random variation (corresponding to developmental noise) in patterning. Specifically, we modified the parameters of simulations corresponding to variation in the LALI-type to generate the full range of phenotypic variation in color pattern seen on the heads of eight leopard geckos. We observed that over the range of these parameters, variation in color pattern due to LALI-type variation exceeds that due to developmental noise in the studied gecko cohort. However, the effect of developmental noise on patterning is also substantial. Our approach addresses one of the major goals of evolutionary biology: to quantify the role of stochasticity in shaping phenotypic variation