Accuracy and response-time distributions for decision-making: linear perfect integrators versus nonlinear attractor-based neural circuits

Animals choose actions based on imperfect, ambiguous data. “Noise” inherent in neural processing adds further variability to this already-noisy input signal. Mathematical analysis has suggested that the optimal apparatus (in terms of the speed/accuracy trade-off) for reaching decisions about such noisy inputs is perfect accumulation of the inputs by a temporal integrator. Thus, most highly cited models of neural circuitry underlying decision-making have been instantiations of a perfect integrator. Here, in accordance with a growing mathematical and empirical literature, we describe circumstances in which perfect integration is rendered suboptimal. In particular we highlight the impact of three biological constraints: (1) significant noise arising within the decision-making circuitry itself; (2) bounding of integration by maximal neural firing rates; and (3) time limitations on making a decision. Under conditions (1) and (2), an attractor system with stable attractor states can easily best an integrator when accuracy is more important than speed. Moreover, under conditions in which such stable attractor networks do not best the perfect integrator, a system with unstable initial states can do so if readout of the system’s final state is imperfect. Ubiquitously, an attractor system with a nonselective time-dependent input current is both more accurate and more robust to imprecise tuning of parameters than an integrator with such input. Given that neural responses that switch stochastically between discrete states can “masquerade” as integration in single-neuron and trial-averaged data, our results suggest that such networks should be considered as plausible alternatives to the integrator model.     

Constraints Shape Cell Function and Morphology by Canalizing the Developmental Path along the Waddington's Landscape

Studies performed in absence of gravitational constraint show that a living system is unable to choose between two different phenotypes, thus leading cells to segregate into different, alternative stable states. This finding demonstrates that the genotype does not determine by itself the phenotype but requires additional, physical constraints to finalize cell differentiation. Constraints belong to two classes: holonomic (independent of the system's dynamical states, as being established by the space-time geometry of the field) and non-holonomic (modified during those biological processes to which they contribute in shaping). This latter kind of “constraints”, in which dynamics works on the constraint to recreate them, have emerged as critical determinants of self-organizing systems, by manifesting a “closure of constraints.” Overall, the constraints act by harnessing the “randomness” represented by the simultaneous presence of equiprobable events restraining the system within one attractor. These results cast doubt on the mainstream scientific concept and call for a better understanding of causation in cell biology.     

长江口滨海湿地潮间带生态系统的多稳态特征

多稳态现象普遍存在于多种生态系统中,它与生态系统的健康和可持续发展密切相关,已成为生态学研究的热点与难点,但是目前有关滨海湿地生态系统多稳态的形成机制还缺乏深入研究.本文以崇明东滩鸟类自然保护区的潮间带生态系统为研究对象,通过以下内容,开展滨海湿地多稳态研究: 1)通过验证多稳态的判定依据“双峰”和“阈值”特征,证实长江口潮间带生态系统存在多稳态,并确定其稳态类型;2)通过监测潮间带生态系统水动力过程、沉积动力过程以及盐沼植物生长和扩散情况,分析盐沼植被与沉积地貌之间的正反馈作用,进而探讨潮间带生态系统多稳态的形成机制.结果表明: 1)潮间带生态系统的归一化植被指数(NDVI)频度分布存在明显的双峰特征,且盐沼植物成活存在生物量阈值效应,均证实潮间带生态系统存在多稳态,“盐沼”和“光滩”是潮间带生态系统的两种相对稳定状态;2)崇明东滩盐沼前沿的沉积地貌表现出泥沙快速淤积的趋势,显著促进了盐沼植物的生长,盐沼植物与泥沙淤积之间的这种正反馈作用是潮间带生态系统形成多稳态的主要原因;3)盐沼植被扩散格局监测结果在景观尺度上也表明,泥沙淤积作用促进了潮间带生态系统“盐沼”和“光滩”多稳态的形成.本研究既丰富了滨海湿地稳态转换的机理研究,也为我国开展海岸带保护、修复和管理提供了科学依据,具有重要的理论和实践意义.     

Analysis of Discrete Bioregulatory Networks Using Symbolic Steady States

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.     

Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. These are finite oriented graphs where vertices are the genes involved in the biological system of interest and arrows describe their interactions: a positive (resp. negative) arrow from a gene to another represents an activation (resp. inhibition) of the expression of the latter gene by some product of the former. Once such an interaction graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. There mathematical tools, among others, can be of some help. In this paper we discuss a rule proposed by Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential [C. Soulé, Graphic requirements for multistationarity, ComPlexUs 1 (2003) 123-133] or Boolean [E. Rémy, P. Ruet, D. Thieffry, Graphic requirements for multistability and attractive cycles in a boolean dynamical framework, 2005, Preprint] type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models. We then discuss open problems.     

Positive feedback circuits and memory

The concept of regulatory feedback circuit refers to oriented cyclic interactions between elements of a system. There are two classes of circuits, positive and negative, whose properties are in striking contrast. Positive circuits are a prerequisite for the occurrence of multiple steady states (multistationarity), and hence, they are involved in all processes showing hysteresis or memory. Endogenous or exogenous perturbations can lead the system to exhibit or to evoke one particular stable regime. The role of positive circuits in cell differentiation and in immunology is well documented. Negative circuits are involved in homeostatic regulation, with or without oscillations. The aim of this paper is to show: a) that positive circuits account for many features of memory stricto sensu (i.e., neural memory and mnesic evocation) as well as largo sensu (e.g. differentiation or immunological memory); and b) that simple combinations of positive and negative circuits provide powerful regulatory modules, which can also be associated in batteries. These entities have vast dynamical possibilities in the field of neurobiology, as well as in the fields of differentiation and immunology. Here we consider a universal minimal regulatory module, for which we suggest to adopt the term 'logical regulon', which can be considered as an atom of Jacob's integron. It comprises a positive and a negative circuit in its interaction matrix, and we recall the main results related to the simultaneous presence of these circuits. Finally, we give three applications of this type of interaction matrix. The first two deal with the coexistence of multiple stable steady states and periodicity in differentiation and in an immunological system showing hysteretic properties. The third deals with the dual problems of synchronization and desynchronization of a neural model for hippocampus memory evocation processes.     

On the existence and the role of chaotic processes in the nervous system

Chaos theory is a rapidly growing field. As a technical term, “chaos” refers to deterministic but unpredictable processes being sensitively dependent upon initial conditions. Neurobiological models and experimental results are very complicated and some research groups have tried to pursue the “neuronal chaos”. Babloyantz's group has studied the fractal dimension (d) of electroencephalograms (EEG) in various physiological and pathological states. From deep sleep (d=4) to full awakening (d>8), a hierarchy of “strange” attractors paralles the hierarchy of states of consciousness. In epilepsy (petit mal), despite the turbulent aspect of a seizure, the attractor dimension was near to 2. In Creutzfeld-Jacob disease, the regular EEG activity corresponded to an attractor dimension less than the one measured in deep sleep. Is it healthy to be chaotic? An “active desynchronisation” could be favourable to a physiological system. Rapp's group reported variations of fractal dimension according to particular tasks. During a mental arithmetic task, this dimension increased. In another task, a P300 fractal index decreased when a target was identified. It is clear that the EEG is not representing noise. Its underlying dynamics depends on only a few degrees of freedom despite yet it is difficult to compute accurately the relevant parameters. What is the cognitive role of such a chaotic dynamics? Freeman has studied the olfactory bulb in rabbits and rats for 15 years. Multi-electrode recordings of a few mm² showed a chaotic hierarchy from deep anaesthesia to alert state. When an animal identified a previously learned odour, the fractal dimension of the dynamics dropped off (near limit cycles). The chaotic activity corresponding to an alert-and-waiting state seems to be a field of all possibilities and a focused activity corresponds to a reduction of the attractor in state space. For a couple of years, Freeman has developed a model of the olfactory bulb-cortex system. The behaviour of the simple model “without learning” was quite similar to the real behaviour and a model “with learning” is developed. Recently, more and more authors insisted on the importance of the dynamic aspect of nervous functioning in cognitive modelling. Most of the models in the neural-network field are designed to converge to a stable state (fixed point) because such behaviour is easy to understand and to control. However, some theoretical studies in physics try to understand how a chaotic behaviour can emerge from neural networks. Sompolinsky's group showed that a sharp transition from a stable state to a chaotic state occurred in totally interconnected networks depending on the value of one control parameter. Learning in such systems is an open field. In conclusion, chaos does exist in neurophysiological processes. It is neither a kind of noise nor a pathological sign. Its main role could be to provide diversity and flexibility to physiological processes. Could “strange” attractors in nervous system embody mental forms? This is a difficult but fascinating question.     

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.     

From minimal signed circuits to the dynamics of Boolean regulatory networks

It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (involved in homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a nontrivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally minimal. We then apply these results to the vertebrate immune system, and show that the two minimal functional positive circuits of the model indeed behave as modules which combine to explain the presence of the three stable states corresponding to the Th0, Th1 and Th2 cells. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Qualitative dynamics of a network model of regulation of the immune system: A rationale for the IgM to IgG switch

A mathematical model has been developed that simulates some of the main features of a network theory of regulation of the immune system. According to the network viewpoint, the V regions (idiotypes) on antibodies and lymphocytes are self-antigens, to which other lymphocytes of the system can respond specifically, just as they respond to foreign antigens. The resultant couplings between the lymphocytes are considered to be basic for the regulation of the system.The present mathematical model simulates the interactions between cells that recognize the antigen (“positive cells”), and “negative cells” that have receptors that specifically recognize the V regions of the positive cells. The mathematical model incorporates only the interactions that are postulated to be important in the four steady states of the theory, and includes neither the antigen nor any accessory (“A”) cells. The effects of both antigen-specific and anti-idiotypic T and B cells are included, as well as antigen-specific and anti-idiotypic T cell factors, and the two main classes of antibodies. The model is a first order autonomous ordinary differential equation in two variables. We describe a geometric technique that gives strong information on the model, without explicitly solving the ordinary differential equation. This technique proves to be powerful in permitting us to systematically scan the parameter space of the model. The detailed analysis leads to support for the idea that the model provides a rationale for the switch observed in the immune system from the production of one major class of antibody (IgM) to the other major class (IgG). The analysis also leads to a new, previously unsuspected possibility for the nature of the suppressed state within the context of the postulates of the symmetrical network theory.     

The field-dependence of the solid-state photo-CIDNP effect in two states of heliobacterial reaction centers

The solid-state photo-CIDNP (photochemically induced dynamic nuclear polarization) effect is studied in photosynthetic reaction centers of Heliobacillus mobilis at different magnetic fields by ¹³C MAS (magic-angle spinning) NMR spectroscopy. Two active states of heliobacterial reaction centers are probed: an anaerobic preparation of heliochromatophores (“Braunstoff”, German for “brown substance”) as well as a preparation of cells after exposure to oxygen (“Grünstoff”, “green substance”). Braunstoff shows significant increase of enhanced absorptive (positive) signals toward lower magnetic fields, which is interpreted in terms of an enhanced differential relaxation (DR) mechanism. In Grünstoff, the signals remain emissive (negative) at two fields, confirming that the influence of the DR mechanism is comparably low.     

Discrete time piecewise affine models of genetic regulatory networks

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. These models reduce to boolean networks in one limiting case of a parameter, and their asymptotic dynamics approaches that of a differential equation in another limiting case of this parameter. For intermediate values, the model present an original phenomenology which is argued to be due to delay effects. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle – with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.     

A Computational Study of Alternate SELEX

Systematic evolution of ligands by exponential enrichment (SELEX) is a procedure for identifying nucleic acid (NA) molecules with affinities for specific target species, such as proteins, peptides, or small organic molecules. Here, we extend the work in Seo et al. (Bull Math Biol 72:1623–1665, 2010) (multiple-target SELEX or positive SELEX) and examine an alternate SELEX process with multiple targets by incorporating negative selection into a positive SELEX protocol. The alternate SELEX process is done iteratively by alternating several positive selection rounds with several negative selection rounds. At the end of each positive selection round, NAs are eluted from the bound product and amplified by polymerase chain reaction (PCR) to increase the size of the pool of NA species that bind preferentially to the given positive target vector. The enriched population of NAs is then exposed to the negative targets (undesired targets). The free NA species (instead of the bound NA species being eluted) are retained and amplified by PCR (negative selection). The goal is to minimize an enrichment of nonspecifically binding NAs against multiple targets. While positive selection alone results in a pool of NAs that bind tightly to a given target vector, negative selection results in the subset of the NAs that bind best to the nontarget vectors that are also present. By alternating the two processes, we eventually obtain a refined population of nucleic acids that bind to the desired target(s) with high “selectivity” and “specificity.” In the present paper, we give formulations of the negative and alternate selection processes and define their efficiencies in a meaningful way. We study the asymptotic behavior of alternate SELEX system as a discrete-time dynamical system. To do this, we use the chemical potential to examine how alternate SELEX leads to the selection of NAs with more specific interactions when the ratio of the number of positive selection rounds to the number of negative selection rounds is fixed. Alternate SELEX is said to be globally asymptotically stable if, given the initial target vector and a fixed ratio, the distribution of the limiting NA fractions does not depend on the relative concentrations of the NAs in the initial pool (provided that all of the NA species are initially present in the initial pool). We state conditions on the matrix of NA—target affinities that determine when the alternate SELEX process is globally asymptotically stable in this sense and illustrate these results computationally.     

Persistence despite perturbations for interacting populations

Two definitions of persistence despite perturbations in deterministic models are presented. The first definition, persistence despite frequent small perturbations, is shown to be equivalent to the existence of a positive attractor i.e. an attractor bounded away from extinction. The second definition, persistence despite rare large perturbations, is shown to be equivalent to permanence i.e. a positive attractor whose basin of attraction includes all positive states. Both definitions set up a natural dichotomy for classifying models of interacting populations. Namely, a model is either persistent despite perturbations or not. When it is not persistent, it follows that all initial conditions are prone to extinction due to perturbations of the appropriate type. For frequent small perturbations, this method of classification is shown to be generically robust: there is a dense set of models for which persistent (respectively, extinction prone) models lies within an open set of persistent (resp. extinction prone) models. For rare large perturbations, this method of classification is shown not to be generically robust. Namely, work of Josef Hofbauer and the author have shown there are open sets of ecological models containing a dense sets of permanent models and a dense set of extinction prone models. The merits and drawbacks of these different definitions are discussed.     

Reprogramming cell fates: reconciling rarity with robustness

The stunning possibility of “reprogramming” differentiated somatic cells to express a pluripotent stem cell phenotype (iPS, induced pluripotent stem cell) and the “ground state” character of pluripotency reveal fundamental features of cell fate regulation that lie beyond existing paradigms. The rarity of reprogramming events appears to contradict the robustness with which the unfathomably complex phenotype of stem cells can reliably be generated. This apparent paradox, however, is naturally explained by the rugged “epigenetic landscape” with valleys representing “preprogrammed” attractor states that emerge from the dynamical constraints of the gene regulatory network. This article provides a pedagogical primer to the fundamental principles of gene regulatory networks as integrated dynamic systems and reviews recent insights in gene expression noise and fate determination, thereby offering a formal framework that may help us to understand why cell fate reprogramming events are inherently rare and yet so robust.     

Carcinogenesis: alterations in reciprocal interactions of normal functional structure of biologic systems

The evolution of biologic systems (BS) includes functional mechanisms that in some conditions may lead to the development of cancer. Using mathematical group theory and matrix analysis, previously, it was shown that normally functioning BS are steady functional structures regulated by three basis regulatory components: reciprocal links (RL), negative feedback (NFB) and positive feedback (PFB). Together, they form an integrative unit maintaining system’s autonomy and functional stability. It is proposed that phylogenetic development of different species is implemented by the splitting of “rudimentary” characters into two relatively independent functional parts that become encoded in chromosomes. The functional correlate of splitting mechanisms is RL. Inversion of phylogenetic mechanisms during ontogenetic development leads cell differentiation until cells reach mature states. Deterioration of reciprocal structure in the genome during ontogenesis gives rise of pathological conditions characterized by unsteadiness of the system. Uncontrollable cell proliferation and invasive cell growth are the leading features of the functional outcomes of malfunctioning systems. The regulatory element responsible for these changes is RL. In matrix language, pathological regulation is represented by matrices having positive values of diagonal elements (TrA?>?0) and also positive values of matrix determinant (detA?>?0). Regulatory structures of that kind can be obtained if the negative entry of the matrix corresponding to RL is replaced with the positive one. To describe not only normal but also pathological states of BS, a unit matrix should be added to the basis matrices representing RL, NFB and PFB. A mathematical structure corresponding to the set of these four basis functional patterns (matrices) is a split quaternion (coquaternion). The structure and specific role of basis elements comprising four-dimensional linear space of split quaternions help to understand what changes in mechanism of cell differentiation may lead to cancer development.     

Are systems with strong underlying abiotic regimes more likely to exhibit alternative stable states?

Suding et al. (2004) demonstrate how conceptual advances in alternative ecosystem states theory have led to a greater understanding of why degraded systems are often resilient to restoration management. In their review they pose one (of several) ‘outstanding’ questions (Box 3 in Suding et al. 2004 ): “Are there predictable characteristics that indicate when a system will follow a successional pathway and/or that indicate the presence or absence of alternative ecosystem states?” We suggest that the persistence of alternative stable states might be predicted from simple consideration of assembly rules for systems structured along a gradient of environmental adversity. We raise the hypothesis that strongly abiotically‐ or disturbance‐structured assemblages, with nonrandom trait under‐dispersion ( Weiher and Keddy 1995 ), are more likely to exhibit catastrophic phase shifts in community structure than assemblages which are weakly structured by environmental adversity.     

Decision-making neural circuits mediating social behaviors

We propose a mathematical model of a continuous attractor network that controls social behaviors. The model is examined with bifurcation analysis and computer simulations. The results show that the model exhibits stable steady states and thresholds for steady state transitions corresponding to some experimentally observed behaviors, such as aggression control. The performance of the model and the relation with experimental evidence are discussed.     

Effects of Time-Dependent Stimuli in a Competitive Neural Network Model of Perceptual Rivalry

We analyze a competitive neural network model of perceptual rivalry that receives time-varying inputs. Time-dependence of inputs can be discrete or smooth. Spike frequency adaptation provides negative feedback that generates network oscillations when inputs are constant in time. Oscillations that resemble perceptual rivalry involve only one population being “ON” at a time, which represents the dominance of a single percept at a time. As shown in Laing and Chow (J. Comput. Neurosci. 12(1):39–53, 2002), for sufficiently high contrast, one can derive relationships between dominance times and contrast that agree with Levelt’s propositions (Levelt in On binocular rivalry, 1965). Time-dependent stimuli give rise to novel network oscillations where both, one, or neither populations are “ON” at any given time. When a single population receives an interrupted stimulus, the fundamental mode of behavior we find is phase-locking, where the temporally driven population locks its state to the stimulus. Other behaviors are analyzed as bifurcations from this forced oscillation, using fast/slow analysis that exploits the slow timescale of adaptation. When both populations receive time-varying input, we find mixtures of fusion and sole population dominance, and we partition parameter space into particular oscillation types. Finally, when a single population’s input contrast is smoothly varied in time, 1:n mode-locked states arise through period-adding bifurcations beyond phase-locking. Our results provide several testable predictions for future psychophysical experiments on perceptual rivalry.