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.     

The modularity of biological regulatory networks

A useful approach to complex regulatory networks consists of modeling their elements and interactions by Boolean equations. In this context, feedback circuits (i.e. circular sequences of interactions) have been shown to play key dynamical roles: whereas positive circuits are able to generate multistationarity, negative circuits may generate oscillatory behavior. In this paper, we principally focus on the case of gene networks. These are represented by fully connected Boolean networks where each element interacts with all elements including itself. Flexibility in network design is introduced by the use of Boolean parameters, one associated with each interaction or group of interactions affecting a given element. Within this formalism, a feedback circuit will generate its typical dynamical behavior (i.e. multistationarity or oscillations) only for appropriate values of some of the logical parameters. Whenever it does, we say that the circuit is 'functional'. More interestingly, this formalism allows the computation of the constraints on the logical parameters to have any feedback circuit functional in a network. Using this methodology, we found that the fraction of the total number of consistent combinations of parameter values that make a circuit functional decreases geometrically with the circuit length. From a biological point of view, this suggests that regulatory networks could be decomposed into small and relatively independent feedback circuits or 'regulatory modules'.     

Reversible plasticity of fear memory-encoding amygdala synaptic circuits even after fear memory consolidation

It is generally believed that after memory consolidation, memory-encoding synaptic circuits are persistently modified and become less plastic. This, however, may hinder the remaining capacity of information storage in a given neural circuit. Here we consider the hypothesis that memory-encoding synaptic circuits still retain reversible plasticity even after memory consolidation. To test this, we employed a protocol of auditory fear conditioning which recruited the vast majority of the thalamic input synaptic circuit to the lateral amygdala (T-LA synaptic circuit; a storage site for fear memory) with fear conditioning-induced synaptic plasticity. Subsequently the fear memory-encoding synaptic circuits were challenged with fear extinction and re-conditioning to determine whether these circuits exhibit reversible plasticity. We found that fear memory-encoding T-LA synaptic circuit exhibited dynamic efficacy changes in tight correlation with fear memory strength even after fear memory consolidation. Initial conditioning or re-conditioning brought T-LA synaptic circuit near the ceiling of their modification range (occluding LTP and enhancing depotentiation in brain slices prepared from conditioned or re-conditioned rats), while extinction reversed this change (reinstating LTP and occluding depotentiation in brain slices prepared from extinguished rats). Consistently, fear conditioning-induced synaptic potentiation at T-LA synapses was functionally reversed by extinction and reinstated by subsequent re-conditioning. These results suggest reversible plasticity of fear memory-encoding circuits even after fear memory consolidation. This reversible plasticity of memory-encoding synapses may be involved in updating the contents of original memory even after memory consolidation.     

Drosophila olfactory memory: single genes to complex neural circuits

A central goal of neuroscience is to understand how neural circuits encode memory and guide behaviour. Studying simple, genetically tractable organisms, such as Drosophila melanogaster, can illuminate principles of neural circuit organization and function. Early genetic dissection of D. melanogaster olfactory memory focused on individual genes and molecules. These molecular tags subsequently revealed key neural circuits for memory. Recent advances in genetic technology have allowed us to manipulate and observe activity in these circuits, and even individual neurons, in live animals. The studies have transformed D. melanogaster from a useful organism for gene discovery to an ideal model to understand neural circuit function in memory.     

Long-term plasticity at inhibitory synapses

Experience-dependent modifications of neural circuits and function are believed to heavily depend on changes in synaptic efficacy such as LTP/LTD. Hence, much effort has been devoted to elucidating the mechanisms underlying these forms of synaptic plasticity. Although most of this work has focused on excitatory synapses, it is now clear that diverse mechanisms of long-term inhibitory plasticity have evolved to provide additional flexibility to neural circuits. By changing the excitatory/inhibitory balance, GABAergic plasticity can regulate excitability, neural circuit function and ultimately, contribute to learning and memory, and neural circuit refinement. Here we discuss recent advancements in our understanding of the mechanisms and functional relevance of GABAergic inhibitory synaptic plasticity.     

Positive and negative feedback: striking a balance between necessary antagonists

Most biological regulation systems comprise feedback circuits as crucial components. Negative feedback circuits have been well understood for a very long time; indeed, their understanding has been the basis for the engineering of cybernetic machines exhibiting stable behaviour. The importance of positive feedback circuits, considered as "vicious circles", has however been underestimated. In this article, we give a demonstration based on degree theory for vector fields of the conjecture, made by René Thomas, that the presence of positive feedback circuits is a necessary condition for autonomous differential systems, covering a wide class of biologically relevant systems, to possess multiple steady states. We also show ways to derive constraints on the weights of positive and negative feedback circuits. These qualitative and quantitative results provide, respectively, structural constraints (i.e. related to the interaction graph) and numerical constraints (i.e. related to the magnitudes of the interactions) on systems exhibiting complex behaviours, and should make it easier to reverse-engineer the interaction networks animating those systems on the basis of partial, sometimes unreliable, experimental data. We illustrate these concepts on a model multistable switch, in the context of cellular differentiation, showing a requirement for sufficient cooperativity. Further developments are expected in the discovery and modelling of regulatory networks in general, and in the interpretation of bio-array hybridization and proteomics experiments in particular.     

Dynamics of chromatic adaptation in cones of freshwater turtle

The closer the wavelength of a steady background of monochromatic light is to the peak sensitivity of a cone that is being illuminated, the stronger is the desensitization of that cone; this is chromatic adaptation. A model of the freshwater turtle retina with the neural components of chromatic adaptation via negative feedback circuits is used to simulate and study various aspects of chromatic adaptation. An internal negative feedback circuit resides solely within the cone pedicle and thereby, its adaptive effects are relatively specific, so that univariance is maintained. The cone-L-horizontal cell circuit is an external negative feedback circuit and its adaptive effects are less specific since all 3 chromatic cone types are involved, so that univariance is violated. Chromatic adaptation is the result of the decrease in the cone gain due to the dependency of the gains of the negative feedback circuits on the mean illuminance level. The results of the model are consistent with von Kries law, but the changes in gains of the cones due to chromatic adaptation are dependent on wavelength, intensity of the adapting light and size.     

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.     

Quantifying dynamic stability of genetic memory circuits

Bistability/Multistability has been found in many biological systems including genetic memory circuits. Proper characterization of system stability helps to understand biological functions and has potential applications in fields such as synthetic biology. Existing methods of analyzing bistability are either qualitative or in a static way. Assuming the circuit is in a steady state, the latter can only reveal the susceptibility of the stability to injected DC noises. However, this can be inappropriate and inadequate as dynamics are crucial for many biological networks. In this paper, we quantitatively characterize the dynamic stability of a genetic conditional memory circuit by developing new dynamic noise margin (DNM) concepts and associated algorithms based on system theory. Taking into account the duration of the noisy perturbation, the DNMs are more general cases of their static counterparts. Using our techniques, we analyze the noise immunity of the memory circuit and derive insights on dynamic hold and write operations. Considering cell-to-cell variations, our parametric analysis reveals that the dynamic stability of the memory circuit has significantly varying sensitivities to underlying biochemical reactions attributable to differences in structure, time scales, and nonlinear interactions between reactions. With proper extensions, our techniques are broadly applicable to other multistable biological systems.     

Multiple forms of activity-dependent competition refine hippocampal circuits in vivo

Efficient memory formation relies on the establishment of functional hippocampal circuits. It has been proposed that synaptic connections are refined by neural activity to form functional brain circuitry. However, it is not known whether and how hippocampal connections are refined by neural activity in vivo. Using a mouse genetic system in which restricted populations of neurons in the hippocampal circuit are inactivated, we show that inactive axons are eliminated after they develop through a competition with active axons. Remarkably, in the dentate gyrus, which undergoes neurogenesis throughout life, axon refinement is achieved by a competition between mature and young neurons. These results demonstrate that activity-dependent competition plays multiple roles in the establishment of functional memory circuits in vivo.     

Correlates of cytotoxic T-lymphocyte-mediated virus control: implications for immunosuppressive infections and their treatment

A very important question in immunology is to determine which factors decide whether an immune response can efficiently clear or control a viral infection, and under what circumstances we observe persistent viral replication and pathology. This paper summarizes how mathematical models help us gain new insights into these questions, and explores the relationship between antiviral therapy and long-term immunological control in human immunodeficiency virus (HIV) infection. We find that cytotoxic T lymphocyte (CTL) memory, defined as antigen-independent persistence of CTL precursors, is necessary for the CTL response to clear an infection. The presence of such a memory response is associated with the coexistence of many CTL clones directed against multiple epitopes. If CTL memory is inefficient, then persistent replication can be established. This outcome is associated with a narrow CTL response directed against only one or a few viral epitopes. If the virus replicates persistently, occurrence of pathology depends on the level of virus load at equilibrium, and this can be determined by the overall efficacy of the CTL response. Mathematical models suggest that controlled replication is reflected by a positive correlation between CTLs and virus load. On the other hand, uncontrolled viral replication results in higher loads and the absence of a correlation between CTLs and virus load. A negative correlation between CTLs and virus load indicates that the virus actively impairs immunity, as observed with HIV. Mathematical models and experimental data suggest that HIV persistence and pathology are caused by the absence of sufficient CTL memory. We show how mathematical models can help us devise therapy regimens that can restore CTL memory in HIV patients and result in long-term immunological control of the virus in the absence of life-long treatment.     

Primal-size neural circuits in meta-periodic interaction

Experimental observations of simultaneous activity in large cortical areas have seemed to justify a large network approach in early studies of neural information codes and memory capacity. This approach has overlooked, however, the segregated nature of cortical structure and functionality. Employing graph-theoretic results, we show that, given the estimated number of neurons in the human brain, there are only a few primal sizes that can be attributed to neural circuits under probabilistically sparse connectivity. The significance of this finding is that neural circuits of relatively small primal sizes in cyclic interaction, implied by inhibitory interneuron potentiation and excitatory inter-circuit potentiation, generate relatively long non-repetitious sequences of asynchronous primal-length periods. The meta-periodic nature of such circuit interaction translates into meta-periodic firing-rate dynamics, representing cortical information. It is finally shown that interacting neural circuits of primal sizes 7 or less exhaust most of the capacity of the human brain, with relatively little room to spare for circuits of larger primal sizes. This also appears to ratify experimental findings on the human working memory capacity.     

Affective empathy and prosocial behavior in rodents

Empathy is an essential function for humans as social animals. Emotional contagion, the basic form of afffective empathy, comprises the cognitive process of perceiving and sharing the affective state of others. The observational fear assay, an animal model of emotional contagion, has enabled researchers to undertake molecular, cellular, and circuit mechanism of this behavior. Such studies have revealed that observational fear is mediated through neural circuits involved in processing the affective dimension of direct pain experiences. A mouse can also respond to milder social stimuli induced by either positive or negative emotional changes in another mouse, which seems not dependent on the affective pain circuits. Further studies should explore how different neural circuits contribute to integrating different dimensions of affective empathy.     

A description of dynamical graphs associated to elementary regulatory circuits

The biological and dynamical importance of feedback circuits in regulatory graphs has often been emphasized. The work presented here aims at completely describing the dynamics of isolated elementary regulatory circuits. Our analytical approach is based on a discrete formal framework, built upon the logical approach of R. Thomas. Given a regulatory circuit, we show that the structure of synchronous and asynchronous dynamical graphs depends only on the length of the circuit (number of genes) and on its sign (which depends on the parity of the number of negative interactions). This work constitutes a first step towards the analytical characterisation of discrete dynamical graphs for more complex regulatory networks in terms of contributions corresponding to their embedded elementary circuits.     

Addiction: Decreased reward sensitivity and increased expectation sensitivity conspire to overwhelm the brain's control circuit

Based on brain imaging findings, we present a model according to which addiction emerges as an imbalance in the information processing and integration among various brain circuits and functions. The dysfunctions reflect (a) decreased sensitivity of reward circuits, (b) enhanced sensitivity of memory circuits to conditioned expectations to drugs and drug cues, stress reactivity, and (c) negative mood, and a weakened control circuit. Although initial experimentation with a drug of abuse is largely a voluntary behavior, continued drug use can eventually impair neuronal circuits in the brain that are involved in free will, turning drug use into an automatic compulsive behavior. The ability of addictive drugs to co‐opt neurotransmitter signals between neurons (including dopamine, glutamate, and GABA) modifies the function of different neuronal circuits, which begin to falter at different stages of an addiction trajectory. Upon exposure to the drug, drug cues or stress this results in unrestrained hyperactivation of the motivation/drive circuit that results in the compulsive drug intake that characterizes addiction.