Classification of infinitesimal homeostasis in four-node input-output networks.

An input-output network has an input node [Formula: see text], an output node o, and regulatory nodes [Formula: see text]. Such a network is a core network if each [Formula: see text] is downstream from [Formula: see text] and upstream from o. Wang et al. (J Math Biol 82:62, 2021. https://doi.org/10.1007/s00285-021-01614-1 ) show that infinitesimal homeostasis can be classified in biochemical networks through infinitesimal homeostasis in core subnetworks. Golubitsky and Wang (J Math Biol 10:1-23, 2020) show that there are three types of 3-node core networks and three types of infinitesimal homeostasis in 3-node core networks. This paper uses the theory developed in Wang et al. (2021) to show that there are twenty types of 4-node core networks (Theorem 1.3) and seventeen types of infinitesimal homeostasis in 4-node core networks (Theorem 1.7). Biological contexts illustrate the classification theorems and show that the theory can be an aid when calculating homeostasis in specific biochemical networks.

The structure of infinitesimal homeostasis in input-output networks.

Homeostasis refers to a phenomenon whereby the output [Formula: see text] of a system is approximately constant on variation of an input [Formula: see text]. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs [Formula: see text] with a distinguished input node [Formula: see text], a different distinguished output node o, and a number of regulatory nodes [Formula: see text]. In these models the input-output map [Formula: see text] is defined by a stable equilibrium [Formula: see text] at [Formula: see text]. Stability implies that there is a stable equilibrium [Formula: see text] for each [Formula: see text] near [Formula: see text] and infinitesimal homeostasis occurs at [Formula: see text] when [Formula: see text]. We show that there is an [Formula: see text] homeostasis matrix [Formula: see text] for which [Formula: see text] if and only if [Formula: see text]. We note that the entries in H are linearized couplings and [Formula: see text] is a homogeneous polynomial of degree [Formula: see text] in these entries. We use combinatorial matrix theory to factor the polynomial [Formula: see text] and thereby determine a menu of different types of possible homeostasis associated with each digraph [Formula: see text]. Specifically, we prove that each factor corresponds to a subnetwork of [Formula: see text]. The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of [Formula: see text] without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of [Formula: see text]. There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).

Infinitesimal homeostasis in three-node input-output networks.

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable [Formula: see text]. Homeostasis plays an important role in the regulation of biological systems, cf.  Ferrell (Cell Syst 2:62-67, 2016), Tang and McMillen (J Theor Biol 408:274-289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to [Formula: see text] is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387-407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input-output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1-24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input-output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input-output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760-773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305-364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264-275, 2014. https://doi.org/10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104-110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma et al. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities.

Symmetry of generalized rivalry network models determines patterns of interocular grouping in four-location binocular rivalry.

Previously, symmetry of network models has been proposed to account for interocular grouping during binocular rivalry. Here, we construct and analyze generalized rivalry network models with different types of symmetry (based on different kinds of excitatory coupling) to derive predictions of possible perceptual states in 12 experiments with four retinal locations. Percepts in binocular rivalry involving more than three locations have not been empirically investigated due to the difficulty in reporting simultaneous percepts at multiple locations. Here, we develop a novel reporting procedure in which the stimulus disappears when the subject is cued to report the simultaneously perceived colors in all four retinal locations. This procedure ensures that simultaneous rather than sequential percepts are reported. The procedure was applied in 12 experiments with six binocular rivalry stimulus configurations, all consisting of dichoptic displays of red and green squares at four locations. We call configurations with an even or odd number of red squares even or odd configurations, respectively. In experiments using even stimulus configurations, we found that even percepts were more frequently observed than odd percepts, whereas in experiments using odd stimulus configurations even and odd percepts were observed with equal probability. The generalized rivalry network models in which couplings depend on stimulus features and spatial configurations was in better agreement with the empirical results. We conclude that the excitatory coupling strength in the horizontal and vertical configurations are different and the coupling strengths between the same color and between different colors are different.NEW & NOTEWORTHY Wilson network models of interocular groupings during binocular rivalry are constructed by considering features that indicate equal coupling strengths. Network symmetries, based on equal couplings, predict percepts. For a four-location rivalry experiment with red or green squares at each location, we analyze different possible Wilson networks. In our experiments we develop a novel reporting procedure and show that networks in which stimulus features and spatial configurations are distinguished best agree with experiments.

Homeostasis in a feed forward loop gene regulatory motif.

The internal state of a cell is affected by inputs from the extra-cellular environment such as external temperature. If some output, such as the concentration of a target protein, remains approximately constant as inputs vary, the system exhibits homeostasis. Special sub-networks called motifs are unusually common in gene regulatory networks (GRNs), suggesting that they may have a significant biological function. Potentially, one such function is homeostasis. In support of this hypothesis, we show that the feed-forward loop GRN produces homeostasis. Here the inputs are subsumed into a single parameter that affects only the first node in the motif, and the output is the concentration of a target protein. The analysis uses the notion of infinitesimal homeostasis, which occurs when the input-output map has a critical point (zero derivative). In model equations such points can be located using implicit differentiation. If the second derivative of the input-output map also vanishes, the critical point is a chair: the output rises roughly linearly, then flattens out (the homeostasis region or plateau), and then starts to rise again. Chair points are a common cause of homeostasis. In more complicated equations or networks, numerical exploration would have to augment analysis. Thus, in terms of finding chairs, this paper presents a proof of concept. We apply this method to a standard family of differential equations modeling the feed-forward loop GRN, and deduce that chair points occur. This function determines the production of a particular mRNA and the resulting chair points are found analytically. The same method can potentially be used to find homeostasis regions in other GRNs. In the discussion and conclusion section, we also discuss why homeostasis in the motif may persist even when the rest of the network is taken into account.

Analysis of Homeostatic Mechanisms in Biochemical Networks.

Cell metabolism is an extremely complicated dynamical system that maintains important cellular functions despite large changes in inputs. This "homeostasis" does not mean that the dynamical system is rigid and fixed. Typically, large changes in external variables cause large changes in some internal variables so that, through various regulatory mechanisms, certain other internal variables (concentrations or velocities) remain approximately constant over a finite range of inputs. Outside that range, the mechanisms cease to function and concentrations change rapidly with changes in inputs. In this paper we analyze four different common biochemical homeostatic mechanisms: feedforward excitation, feedback inhibition, kinetic homeostasis, and parallel inhibition. We show that all four mechanisms can occur in a single biological network, using folate and methionine metabolism as an example. Golubitsky and Stewart have proposed a method to find homeostatic nodes in networks. We show that their method works for two of these mechanisms but not the other two. We discuss the many interesting mathematical and biological questions that emerge from this analysis, and we explain why understanding homeostatic control is crucial for precision medicine.

Dimorphism by Singularity Theory in a Model for River Ecology.

Geritz, Gyllenberg, Jacobs, and Parvinen show that two similar species can coexist only if their strategies are in a sector of parameter space near a nondegenerate evolutionarily singular strategy. We show that the dimorphism region can be more general by using the unfolding theory of Wang and Golubitsky near a degenerate evolutionarily singular strategy. Specifically, we use a PDE model of river species as an example of this approach. Our finding shows that the dimorphism region can exhibit various different forms that are strikingly different from previously known results in adaptive dynamics.

Homeostasis, singularities, and networks.

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter [Formula: see text] varies over some interval. We discuss two main aspects of homeostasis, both related to the effect of coordinate changes on the input-output map. The first is a reformulation of homeostasis in the context of singularity theory, achieved by replacing 'approximately constant over an interval' by 'zero derivative of the output with respect to the input at a point'. Unfolding theory then classifies all small perturbations of the input-output function. In particular, the 'chair' singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding [Formula: see text] is derived and the region of approximate homeostasis is deduced. The results are motivated by data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points in mathematical models by implicit differentiation and apply it to a model of lateral inhibition. The second asks when homeostasis is invariant under appropriate coordinate changes. This is false in general, but for network dynamics there is a natural class of coordinate changes: those that preserve the network structure. We characterize those nodes of a given network for which homeostasis is invariant under such changes. This characterization is determined combinatorially by the network topology.

Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics.

We survey general results relating patterns of synchrony to network topology, applying the formalism of coupled cell systems. We also discuss patterns of phase-locking for periodic states, where cells have identical waveforms but regularly spaced phases. We focus on rigid patterns, which are not changed by small perturbations of the differential equation. Symmetry is one mechanism that creates patterns of synchrony and phase-locking. In general networks, there is another: balanced colorings of the cells. A symmetric network may have anomalous patterns of synchrony and phase-locking that are not consequences of symmetry. We introduce basic notions on coupled cell networks and their associated systems of admissible differential equations. Periodic states also possess spatio-temporal symmetries, leading to phase relations; these are classified by the H/K theorem and its analog for general networks. Systematic general methods for computing the stability of synchronous states exist for symmetric networks, but stability in general networks requires methods adapted to special classes of model equations.

Singularity theory of fitness functions under dimorphism equivalence.

We apply singularity theory to classify monomorphic singular points as they occur in adaptive dynamics. Our approach is based on a new equivalence relation called dimorphism equivalence, which is the largest equivalence relation on strategy functions that preserves ESS singularities, CvSS singularities, and dimorphisms. Specifically, we classify singularities up to topological codimension two and compute their normal forms and universal unfoldings. These calculations lead to the classification of local mutual invasibility plots that can be seen generically in systems with two parameters.

Recent advances in symmetric and network dynamics.

We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts René Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network.

Network symmetry and binocular rivalry experiments.

Hugh Wilson has proposed a class of models that treat higher-level decision making as a competition between patterns coded as levels of a set of attributes in an appropriately defined network (Cortical Mechanisms of Vision, pp. 399-417, 2009; The Constitution of Visual Consciousness: Lessons from Binocular Rivalry, pp. 281-304, 2013). In this paper, we propose that symmetry-breaking Hopf bifurcation from fusion states in suitably modified Wilson networks, which we call rivalry networks, can be used in an algorithmic way to explain the surprising percepts that have been observed in a number of binocular rivalry experiments. These rivalry networks modify and extend Wilson networks by permitting different kinds of attributes and different types of coupling. We apply this algorithm to psychophysics experiments discussed by Kovács et al. (Proc. Natl. Acad. Sci. USA 93:15508-15511, 1996), Shevell and Hong (Vis. Neurosci. 23:561-566, 2006; Vis. Neurosci. 25:355-360, 2008), and Suzuki and Grabowecky (Neuron 36:143-157, 2002). We also analyze an experiment with four colored dots (a simplified version of a 24-dot experiment performed by Kovács), and a three-dot analog of the four-dot experiment. Our algorithm predicts surprising differences between the three- and four-dot experiments.

Derived patterns in binocular rivalry networks.

Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Wilson proposed a class of neuronal network models that generalize rivalry to multiple competing patterns. The networks are assumed to have learned several patterns, and rivalry is identified with time periodic states that have periods of dominance of different patterns. Here, we show that these networks can also support patterns that were not learned, which we call derived. This is important because there is evidence for perception of derived patterns in the binocular rivalry experiments of Kovács, Papathomas, Yang, and Fehér. We construct modified Wilson networks for these experiments and use symmetry breaking to make predictions regarding states that a subject might perceive. Specifically, we modify the networks to include lateral coupling, which is inspired by the known structure of the primary visual cortex. The modified network models make expected the surprising outcomes observed in these experiments.

Central pattern generators for bipedal locomotion.

Golubitsky, Stewart, Buono and Collins proposed two models for the achitecture of central pattern generators (CPGs): one for bipeds (which we call leg) and one for quadrupeds (which we call quad). In this paper we use symmetry techniques to classify the possible spatiotemporal symmetries of periodic solutions that can exist in leg (there are 10 nontrivial types) and we explore the possibility that coordinated arm/leg rhythms can be understood, on the CPG level, by a small breaking of the symmetry in quad, which leads to a third CPG architecture arm. Rhythms produced by leg correspond to the bipedal gaits of walk, run, two-legged hop, two-legged jump, skip, gallop, asymmetric hop, and one-legged hop. We show that breaking the symmetry between fore and hind limbs in quad, which yields the CPG arm, leads to periodic solution types whose associated leg rhythms correspond to seven of the eight leg gaits found in leg; the missing biped gait is the asymmetric hop. However, when arm/leg coordination rhythms are considered, we find the correct rhythms only for the biped gaits of two-legged hop, run, and gallop. In particular, the biped gait walk, along with its arm rhythms, cannot be obtained by a small breaking of symmetry of any quadruped gait supported by quad.

What geometric visual hallucinations tell us about the visual cortex.

Many observers see geometric visual hallucinations after taking hallucinogens such as LSD, cannabis, mescaline or psilocybin; on viewing bright flickering lights; on waking up or falling asleep; in "near-death" experiences; and in many other syndromes. Klüver organized the images into four groups called form constants: (I) tunnels and funnels, (II) spirals, (III) lattices, including honeycombs and triangles, and (IV) cobwebs. In most cases, the images are seen in both eyes and move with them. We interpret this to mean that they are generated in the brain. Here, we summarize a theory of their origin in visual cortex (area V1), based on the assumption that the form of the retino-cortical map and the architecture of V1 determine their geometry. (A much longer and more detailed mathematical version has been published in Philosophical Transactions of the Royal Society B, 356 [2001].) We model V1 as the continuum limit of a lattice of interconnected hypercolumns, each comprising a number of interconnected iso-orientation columns. Based on anatomical evidence, we assume that the lateral connectivity between hypercolumns exhibits symmetries, rendering it invariant under the action of the Euclidean group E(2), composed of reflections and translations in the plane, and a (novel) shift-twist action. Using this symmetry, we show that the various patterns of activity that spontaneously emerge when V1's spatially uniform resting state becomes unstable correspond to the form constants when transformed to the visual field using the retino-cortical map. The results are sensitive to the detailed specification of the lateral connectivity and suggest that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges, contours, surfaces, and textures.

Symmetric patterns in linear arrays of coupled cells.

In this note we show how to find patterned solutions in linear arrays of coupled cells. The solutions are found by embedding the system in a circular array with twice the number of cells. The individual cells have a unique steady state, so that the patterned solutions represent a discrete analog of Turing structures in continuous media. We then use the symmetry of the circular array (and bifurcation from an invariant equilibrium) to identify symmetric solutions of the circular array that restrict to solutions of the original linear array. We apply these abstract results to a system of coupled Brusselators to prove that patterned solutions exist. In addition, we show, in certain instances, that these patterned solutions can be found by numerical integration and hence are presumably asymptotically stable.