Emergence of complex behaviour from simple circuit structures

The set of (feedback) circuits of a complex system is the machinery that allows the system to be aware of the levels of its crucial constituents. Circuits can be identified without ambiguity from the elements of the Jacobian matrix of the system. There are two types of circuits: positive if they comprise an even number of negative interactions, negative if this number is odd. The two types of circuits play deeply different roles: negative circuits are required for homeostasis, with or without oscillations, positive circuits are required for multistationarity, and hence, in biology, for differentiation and memory. In non-linear systems, a circuit can positive or negative (an 'ambiguous circuit', depending on the location in phase space. Full circuits are those circuits (or unions of disjoint circuits) that imply all the variables of the system. There is a tight relation between circuits and steady states. Each full circuit, if isolated, generates steady state(s) whose nature (eigenvalues) is determined by the structure of the circuit. Multistationarity requires the presence of at least two full circuits of opposite Eisenfeld signs, or else, an ambiguous circuit. We show how a significant part of the dynamical behaviour of a system can be predicted by a mere examination of its Jacobian matrix. We also show how extremely complex dynamics can be generated by such simple logical structures as a single (full and ambiguous) circuit.     

Invertebrate central pattern generator circuits

There are now a reasonable number of invertebrate central pattern generator (CPG) circuits described in sufficient detail that a mechanistic explanation of how they work is possible. These small circuits represent the best-understood neural circuits with which to investigate how cell-to-cell synaptic connections and individual channel conductances combine to generate rhythmic and patterned output. In this review, some of the main lessons that have appeared from this analysis are discussed and concrete examples of circuits ranging from single phase to multiple phase patterns are described. While it is clear that the cellular components of any CPG are basically the same, the topology of the circuits have evolved independently to meet the particular motor requirements of each individual organism and only a few general principles of circuit operation have emerged. The principal usefulness of small systems in relation to the brain is to demonstrate in detail how cellular infrastructure can be used to generate rhythmicity and form specialized patterns in a way that may suggest how similar processes might occur in more complex systems. But some of the problems and challenges associated with applying data from invertebrate preparations to the brain are also discussed. Finally, I discuss why it is useful to have well-defined circuits with which to examine various computational models that can be validated experimentally and possibly applied to brain circuits when the details of such circuits become available.     

Fading memory and kernel properties of generic cortical microcircuit models

It is quite difficult to construct circuits of spiking neurons that can carry out complex computational tasks. On the other hand even randomly connected circuits of spiking neurons can in principle be used for complex computational tasks such as time-warp invariant speech recognition. This is possible because such circuits have an inherent tendency to integrate incoming information in such a way that simple linear readouts can be trained to transform the current circuit activity into the target output for a very large number of computational tasks. Consequently we propose to analyze circuits of spiking neurons in terms of their roles as analog fading memory and non-linear kernels, rather than as implementations of specific computational operations and algorithms. This article is a sequel to [W. Maass, T. Natschl?ger, H. Markram, Real-time computing without stable states: a new framework for neural computation based on perturbations, Neural Comput. 14 (11) (2002) 2531-2560, Online available as #130 from: ], and contains new results about the performance of generic neural microcircuit models for the recognition of speech that is subject to linear and non-linear time-warps, as well as for computations on time-varying firing rates. These computations rely, apart from general properties of generic neural microcircuit models, just on capabilities of simple linear readouts trained by linear regression. This article also provides detailed data on the fading memory property of generic neural microcircuit models, and a quick review of other new results on the computational power of such circuits of spiking neurons.     

Reduced models for binocular rivalry

Binocular rivalry occurs when two very different images are presented to the two eyes, but a subject perceives only one image at a given time. A number of computational models for binocular rivalry have been proposed; most can be categorised as either “rate” models, containing a small number of variables, or as more biophysically-realistic “spiking neuron” models. However, a principled derivation of a reduced model from a spiking model is lacking. We present two such derivations, one heuristic and a second using recently-developed data-mining techniques to extract a small number of “macroscopic” variables from the results of a spiking neuron model simulation. We also consider bifurcations that can occur as parameters are varied, and the role of noise in such systems. Our methods are applicable to a number of other models of interest.     

Synthetic circuits reveal how mechanisms of gene regulatory networks constrain evolution

Phenotypic variation is the raw material of adaptive Darwinian evolution. The phenotypic variation found in organismal development is biased towards certain phenotypes, but the molecular mechanisms behind such biases are still poorly understood. Gene regulatory networks have been proposed as one cause of constrained phenotypic variation. However, most pertinent evidence is theoretical rather than experimental. Here, we study evolutionary biases in two synthetic gene regulatory circuits expressed in Escherichia coli that produce a gene expression stripe—a pivotal pattern in embryonic development. The two parental circuits produce the same phenotype, but create it through different regulatory mechanisms. We show that mutations cause distinct novel phenotypes in the two networks and use a combination of experimental measurements, mathematical modelling and DNA sequencing to understand why mutations bring forth only some but not other novel gene expression phenotypes. Our results reveal that the regulatory mechanisms of networks restrict the possible phenotypic variation upon mutation. Consequently, seemingly equivalent networks can indeed be distinct in how they constrain the outcome of further evolution.     

Methods for and results from the study of design principles in molecular systems

Most aspects of molecular biology can be understood in terms of biological design principles. These principles can be loosely defined as qualitative and quantitative features that emerge in evolution and recur more frequently than one would expect by chance alone in biological systems that perform a given type of process or function. Furthermore, such recurrence can be rationalized in terms of the functional advantage that the design provides to the system when compared with possible alternatives. This paper focuses on those design features that can be related to improved functional effectiveness of molecular and regulatory networks. We begin by reviewing assumptions and methods that underlie the study of such principles in molecular networks. We follow by discussing many of the design principles that have been found in genetic, metabolic, and signal transduction circuits. We concentrate mainly on results in the context of Biochemical Systems Theory, although we also briefly discuss other work. We conclude by discussing the importance of these principles for both, understanding the natural evolution of complex networks at the molecular level and for creating artificial biological systems with specific features.     

Relevance of multiple spatial scales in habitat models: A case study with amphibians and grasshoppers

Habitat models for animal species are important tools in conservation planning. We assessed the need to consider several scales in a case study for three amphibian and two grasshopper species in the post-mining landscapes near Leipzig (Germany). The two species groups were selected because habitat analyses for grasshoppers are usually conducted on one scale only whereas amphibians are thought to depend on more than one spatial scale.First, we analysed how the preference to single habitat variables changed across nested scales. Most environmental variables were only significant for a habitat model on one or two scales, with the smallest scale being particularly important. On larger scales, other variables became significant, which cannot be recognized on lower scales. Similar preferences across scales occurred in only 13 out of 79 cases and in 3 out of 79 cases the preference and avoidance for the same variable were even reversed among scales.Second, we developed habitat models by using a logistic regression on every scale and for all combinations of scales and analysed how the quality of habitat models changed with the scales considered. To achieve a sufficient accuracy of the habitat models with a minimum number of variables, at least two scales were required for all species except for Bufo viridis, for which a single scale, the microscale, was sufficient. Only for the European tree frog (Hyla arborea), at least three scales were required.The results indicate that the quality of habitat models increases with the number of surveyed variables and with the number of scales, but costs increase too. Searching for simplifications in multi-scaled habitat models, we suggest that 2 or 3 scales should be a suitable trade-off, when attempting to define a suitable microscale.     

Growth curve models of repeated binary response

Experimental designs that include repeated measures of binary response variables over time and under different conditions are common in biology. In such settings, it is often desirable to characterize the response pattern over time. When response variables are continuous, this characterization can be made in terms of a growth model such as the Potthoff-Roy growth curve model. We illustrate how a similar growth curve modeling strategy can be implemented using weighted least squares (WLS) methods for binary response data. The growth models are constructed in terms of polynomial functions across marginal response. However, when growth models are fit to repeated binary response, the nonsignificant higher-order polynomial functions are dropped from the model, rather than used as covariates. Dropping the nonsignificant polynomials from the model will reduce the number of response functions, and help avoid small-sample problems that can occur when the number of correlated response functions is large and sample sizes are small. The reduced set of response functions are then modeled using WLS methods. We illustrate such models with an example of binary fly oviposition response (accept or reject) exhibited by two populations of flies at four ages to two types of fruit.     

A new necessary condition on interaction graphs for multistationarity

We consider a dynamical system, described by a system of ordinary differential equations, and the associated interaction graphs, which are defined using the matrix of signs of the Jacobian matrix. After stating a few conjectures about the role of circuits in these graphs, we prove two new results relating them to the dynamic behaviour of the system: a sufficient condition for qualitative unstability, and a necessary condition for the existence of several stationary states. These results are illustrated by examples of regulatory modules in two variables, such as those occurring in biological networks.     

From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis

This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. An analysis of the complexity of the algorithm is performed. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster.     

Between-group analysis of microarray data

MOTIVATION: Most supervised classification methods are limited by the requirement for more cases than variables. In microarray data the number of variables (genes) far exceeds the number of cases (arrays), and thus filtering and pre-selection of genes is required. We describe the application of Between Group Analysis (BGA) to the analysis of microarray data. A feature of BGA is that it can be used when the number of variables (genes) exceeds the number of cases (arrays). BGA is based on carrying out an ordination of groups of samples, using a standard method such as Correspondence Analysis (COA), rather than an ordination of the individual microarray samples. As such, it can be viewed as a method of carrying out COA with grouped data. RESULTS: We illustrate the power of the method using two cancer data sets. In both cases, we can quickly and accurately classify test samples from any number of specified a priori groups and identify the genes which characterize these groups. We obtained very high rates of correct classification, as determined by jack-knife or validation experiments with training and test sets. The results are comparable to those from other methods in terms of accuracy but the power and flexibility of BGA make it an especially attractive method for the analysis of microarray cancer data.     

Neuronal circuits on a chip for biological network monitoring

Cultured neuronal networks (CNNs) are a robust model to closely investigate neuronal circuits’ formation and monitor their structural properties evolution. Typically, neurons are cultured in plastic plates or, more recently, in microfluidic platforms with potentially a wide variety of neuroscience applications. As a biological protocol, cell culture integration with a microfluidic system provides benefits such as accurate control of cell seeding area, culture medium renewal, or lower exposure to contamination. The objective of this report is to present a novel neuronal network on a chip device, including a chamber, fabricated from PDMS, vinyl and glass connected to a microfluidic platform to perfuse the continuous flow of culture medium. Network growth is compared in chips and traditional Petri dishes to validate the microfluidic chip performance. The network assessment is performed by computing relevant topological measures like the number of connected neurons, the clustering coefficient, and the shortest path between any pair of neurons throughout the culture's life. The results demonstrate that neuronal circuits on a chip have a more stable network structure and lifespan than developing in conventional settings, and therefore this setup is an advantageous alternative to current culture methods. This technology could lead to challenging applications such as batch drug testing of in vitro cell culture models. From the engineering perspective, a device's advantage is the chance to develop custom designs more efficiently than other microfluidic systems.     

Dominant ionic mechanisms explored in spiking and bursting using local low-dimensional reductions of a biophysically realistic model neuron

The large number of variables involved in many biophysical models can conceal potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them as parameters are varied. To address this issue, we extend a novel model reduction method, based on “scales of dominance,” to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. We divide trajectories into intervals dominated by a smaller number of variables, resulting in a locally reduced hybrid model whose dimension varies between two and six in different temporal regimes. The reduced model exhibits the same modes of oscillation as the 16 dimensional model over a comparable parameter range, and requires fewer ad hoc simplifications than a more traditional reduction to a single, globally valid model. The hybrid model highlights low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze the interactions between different modes of activity.     

Circuits constructed from identified Aplysia neurons exhibit multiple patterns of persistent activity.

We have used identified neurons from the abdominal ganglion of the mollusc Aplysia to construct and analyze two circuits in vitro. Each of these circuits was capable of producing two patterns of persistent activity; that is, they had bistable output states. The output could be switched between the stable states by a brief, external input. One circuit consisted of cocultured L10 and left upper quadrant (LUQ) neurons that formed reciprocal, inhibitory connections. In one stable state L10 was active and the LUQ was quiescent, whereas in the other stable state L10 was quiescent and the LUQ was active. A second circuit consisted of co-cultured L7 and L12 neurons that formed reciprocal, excitatory connections. In this circuit, both cells were quiescent in one stable state and both cells fired continuously in the other state. Bistable output in both circuits resulted from the nonlinear firing characteristics of each neuron and the feedback between the two neurons. We explored how the stability of the neuronal output could be controlled by the background currents injected into each neuron. We observed a relatively well-defined range of currents for which bistability occurred, consistent with the values expected from the measured strengths of the connections and a simple model. Outside of the range, the output was stable in only a single state. These results suggest how stable patterns of output are produced by some in vivo circuits and how command neurons from higher neural centers may control the activity of these circuits. The criteria that guided us in forming our circuits in culture were derived from theoretical studies on the properties of certain neuronal network models (e.g., Hopfield, J. J. 1984. Proc. Natl. Acad. Sci. USA. 81:3088-3092). Our results show that circuits consisting of only two co-cultured neurons can exhibit bistable output states of the form hypothesized to occur in populations of neurons.     

Integrated circuits and molecular components for stress and feeding: implications for eating disorders

Eating disorders are complex brain disorders that afflict millions of individuals worldwide. The etiology of these diseases is not fully understood, but a growing body of literature suggests that stress and anxiety may play a critical role in their development. As our understanding of the genetic and environmental factors that contribute to disease in clinical populations like anorexia nervosa, bulimia nervosa and binge eating disorder continue to grow, neuroscientists are using animal models to understand the neurobiology of stress and feeding. We hypothesize that eating disorder clinical phenotypes may result from stress‐induced maladaptive alterations in neural circuits that regulate feeding, and that these circuits can be neurochemically isolated using animal model of eating disorders.     

Can niche‐based distribution models outperform spatial interpolation?

Aim Distribution modelling relates sparse data on species occurrence or abundance to environmental information to predict the population of a species at any point in space. Recently, the importance of spatial autocorrelation in distributions has been recognized. Spatial autocorrelation can be categorized as exogenous (stemming from autocorrelation in the underlying variables) or endogenous (stemming from activities of the organism itself, such as dispersal). Typically, one asks whether spatial models explain additional variability (endogenous) in comparison to a fully specified habitat model. We turned this question around and asked: can habitat models explain additional variation when spatial structure is accounted for in a fully specified spatially explicit model? The aim was to find out to what degree habitat models may be inadvertently capturing spatial structure rather than true explanatory mechanisms. Location We used data from 190 species of the North American Breeding Bird Survey covering the conterminous United States and southern Canada. Methods We built 13 different models on 190 bird species using regression trees. Our habitat‐based models used climate and landcover variables as independent variables. We also used random variables and simulated ranges to validate our results. The two spatially explicit models included only geographical coordinates or a contagion term as independent variables. As another angle on the question of mechanism vs. spatial structure we pitted a model using related bird species as predictors against a model using randomly selected bird species. Results The spatially explicit models outperformed the traditional habitat models and the random predictor species outperformed the related predictor species. In addition, environmental variables produced a substantial R² in predicting artificial ranges. Main conclusions We conclude that many explanatory variables with suitable spatial structure can work well in species distribution models. The predictive power of environmental variables is not necessarily mechanistic, and spatial interpolation can outperform environmental explanatory variables.     

A spectrum of modularity in multi‐functional gene circuits

A major challenge in systems biology is to understand the relationship between a circuit's structure and its function, but how is this relationship affected if the circuit must perform multiple distinct functions within the same organism? In particular, to what extent do multi‐functional circuits contain modules which reflect the different functions? Here, we computationally survey a range of bi‐functional circuits which show no simple structural modularity: They can switch between two qualitatively distinct functions, while both functions depend on all genes of the circuit. Our analysis reveals two distinct classes: hybrid circuits which overlay two simpler mono‐functional sub‐circuits within their circuitry, and emergent circuits, which do not. In this second class, the bi‐functionality emerges from more complex designs which are not fully decomposable into distinct modules and are consequently less intuitive to predict or understand. These non‐intuitive emergent circuits are just as robust as their hybrid counterparts, and we therefore suggest that the common bias toward studying modular systems may hinder our understanding of real biological circuits.     

The transferability of distribution models across regions: an amphibian case study

Aim Predicting species distribution is of fundamental importance for ecology and conservation. However, distribution models are usually established for only one region and it is unknown whether they can be transferred to other geographical regions. We studied the distribution of six amphibian species in five regions to address the question of whether the effect of landscape variables varied among regions. We analysed the effect of 10 variables extracted in six concentric buffers (from 100 m to 3 km) describing landscape composition around breeding ponds at different spatial scales. We used data on the occurrence of amphibian species in a total of 655 breeding ponds. We accounted for proximity to neighbouring populations by including a connectivity index to our models. We used logistic regression and information‐theoretic model selection to evaluate candidate models for each species. Location Switzerland. Results The explained deviance of each species’ best models varied between 5% and 32%. Models that included interactions between a region and a landscape variable were always included in the most parsimonious models. For all species, models including region‐by‐landscape interactions had similar support (Akaike weights) as models that did not include interaction terms. The spatial scale at which landscape variables affected species distribution varied from 100 m to 1000 m, which was in agreement with several recent studies suggesting that land use far away from the ponds can affect pond occupancy. Main conclusions Different species are affected by different landscape variables at different spatial scales and these effects may vary geographically, resulting in a generally low transferability of distribution models across regions. We also found that connectivity seems generally more important than landscape variables. This suggests that metapopulation processes may play a more important role in species distribution than habitat characteristics.