Homeostasis and Network Invariants

Transcription

1 Homeostasis and Network Invariants Martin Golubitsky Mathematical Biosciences Institute 364 Jennings Hall Ohio State University Columbus OH 43210, USA Ian Stewart Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom June 17, 2015 Abstract Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter varies over some region. We reformulate homeostasis in the context of singularity theory by replacing approximately constant over an interval by zero derivative with respect to the input at a point. General coordinate changes need not leave this condition invariant, but in network dynamics there is a natural class of right network-preserving coordinate changes. We show that these coordinate changes preserve homeostasis in a cell if the cell forms a block in the right core of the network, a combinatorial condition that often occurs. The induced coordinate changes on the singularity are a version of contact equivalence. These results lead to new types of network invariant related to homeostasis. In particular, the chair singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding λ 3 + aλ is derived and the region of approximate homeostasis is deduced. The results are related to data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points by implicit differentiation and discuss chair points in a simple network ODE. Finally we sketch generalizations of the theory to multiparameter systems. 1 Introduction In dynamical systems theory, the main emphasis is on dynamical properties that remain invariant under changes of coordinates. Examples include the topology of the phase portrait, the existence or absence of a chaotic attractor, and the occurrence of a Hopf bifurcation as a parameter is varied. Properties that lack this kind of invariance are to some extent thought of as accidental consequences of a particular choice of coordinates. 1

2 On the other hand, many systems of practical interest involve specific variables with distinguished properties. Applications of dynamical systems theory therefore tread a delicate path, and are often supplemented by model-specific information and analysis. For example Hopf bifurcation might be used to predict the occurrence of a time-periodic state, but model-dependent information is then required to determine the precise waveform of this state. Here we apply a characterization of network-preserving coordinate changes derived in [4] to a question arising in biochemistry: the notion of homeostasis, where at an equilibrium of a system of ODEs some output variable remains approximately constant as an input parameter varies. The dynamical systems viewpoint suggests that a formal definition of homeostasis should be invariant under appropriate coordinate changes. This immediately raises the question: which coordinate changes are appropriate? Clearly they should preserve those qualitative features of the dynamics that are relevant in that context. But they should also preserve the key feature that distinguishes network dynamics from a general dynamical system: the existence of distinguished variables corresponding to the nodes (or cells) of the network. It is therefore natural to require the coordinate changes to preserve the space of admissible vector fields determined by the network topology, [13, 7]. See Section 3. In [4] we characterize five distinct types of coordinate change that preserve the space of admissible vector fields. Only one of these, right changes, is directly relevant to homeostasis for equilibrium states. However, there may be similar applications of other types of coordinate change in particular vector field changes, which preserve the qualitative dynamics to phenomena such as homeostasis in the period or amplitude of an oscillatory state; for example, circadian rhythms. We have not explored this possibility. Throughout we work in the class of fully inhomogenous networks in which all coupling types are distinct (no arrow equivalences between different arrows) and all cells are distinct (no cell equivalences between different cells). This class is natural for biochemical systems where coupling parameters such as reaction rates typically differ from each other. We begin in Section 2 with a biological example of homeostasis in the regulation of the output body temperature in an opossum, when the input environmental temperature varies. The graph of body temperature against environmental temperature λ varies linearly with nonzero slope when λ is either small or large and has a broad flat region that indicates homeostasis in between. This general shape is known as a chair, Nijhout et al. [10, 11] and plays a central role in the paper. This example motivates reformulating homeostasis in terms of the derivative of an output variable with respect to an input being zero at some point and hence being approximately constant near that point. We discuss this mathematical reformulation and consider its invariance under coordinate changes. In particular we remark that in a general dynamical system, the usual coordinate changes can mix up variables and fail to preserve homeostasis. This observation motivates taking network structure into account, because networks have distinguished variables, variables associated to the network nodes. 2

3 Section 3 addresses the network context, recalling the notion of an admissible ODE one that respects the network architecture and providing simple examples to illustrate the idea. In Section 4 we summarize the characterization of a special class of coordinate transformations from [4] that turns out to be appropriate for homeostasis. These are called right network-preserving coordinate changes. We prove that they are precisely the admissible diffeomorphisms for a special subnetwork, the right core, which is defined combinatorially in terms of simple features of the network. An example illustrates how the right core decomposes the cells of the network into blocks for which the tail cell of any input arrow to any cell in the block also lies in the block. We show that blocks are invariant under right network-preserving coordinate changes. We also provide a simple sufficient characterization of networks in which every cell forms a block, so the right network-preserving coordinate changes are independent diffeomorphisms on each cell ( strongly admissible diffeomorphisms [7]). The main theoretical result in the paper occurs in Section 5, where we propose an appropriate class of coordinate changes on the admissible ODE and investigate how it transforms the input-output map, which is defined implicitly by how a block of coordinates of a stable equilibrium of the ODE varies with the input. We prove that when a block consists of a single cell, this type of coordinate change induces, in the language of singularity theory, a version of contact equivalence on the input-output function. Section 6 explores the consequences of this result for homeostasis, with special emphasis on the two simplest singularities: simple homeostasis and the chair. We characterize these singularities and discuss their normal forms (the simplest form into which the singularity can be transformed) and (universal) unfoldings, which classify all small perturbations as other system parameters vary. We examine the unfolding of the chair to estimate the region over which homeostasis occurs, in the sense that the output varies by no more than a small amount δ as the input changes. We relate the unfolding of the chair to observational data on two species of opossum and the spiny rat. Section 7 provides a brief discussion of how chair points can be calculated analytically by implicit differentiation. Section 8 discusses a simple made up model with three cells, working out the right core and finding a chair point. Finally, Section 9 discusses how the methods of the paper might be extended to more general contexts, leading to singularitytheoretic characterizations of more general forms of homeostasis and related effects. 2 Homeostasis Homeostasis occurs when some feature of a system remains essentially constant as an input parameter varies within some range of values. For example, the body temperature of an organism might remain roughly constant despite variations in its environment. (See Figure 1 for such data in the brown opossum where body temperature remains approximately constant over a range of 18 C in environmental temperature [9, 10].) Or in a biochemical network the equilibrium concentration of some important biochemical molecule might not change much when the organism has ingested food. 3

4 Figure 1: Experimental data indicating thermoregulatory homeostasis in the brown opossum. The horizontal axis is environmental temperature ( C) and the vertical axis is body temperature ( C). [9, 10] Homeostasis is almost exactly opposite to bifurcation. At a bifurcation, the state of the system undergoes a change so extensive that some qualitative property (such as number or topological type of attractors) changes. In homeostasis, the state concerned not only remains topologically the same: some feature of that state does not even change quantitatively. For example, if a steady state does not bifurcate as a parameter is varied, that state persists, but can change continuously with the parameter. Homeostasis is stronger: the steady state persists, and in addition some feature of that steady state remains almost constant. Homeostasis is biologically important, because it protects organisms against changes induced by the environment, shortage of some resource, excess of some resource, the effect of ingesting food, as so on. The literature is extensive [14]. However, homeostasis is not merely the existence (and persistence as parameters vary) of a stable equilibrium of the system, for two reasons. First, homeostasis is a stronger condition than the equilibrium varies continuously with parameters, which just states that there is no bifurcation. In the biological context, nonzero linear variation of the equilibrium as parameters change is not normally considered to be homeostasis, unless the slope is very small. For example, in Figure 1, body temperature appears to be varying linearly when the environmental temperature is either below 10 C or above 30 C and is approximately constant in between. Second, some variable(s) of the system may be homeostatic while others undergo larger changes. Indeed, other variables may have to change dramatically to keep some specific variable roughly constant. An example is the regulation of blood sugar concentration by insulin, where insulin levels may have to change substantially in order to keep blood sugar constant. As this example indicates, homeostasis is closely linked to specific variables. Mathematically, we can describe the simplest form of homeostasis as follows. Let x be the variable that is to remain almost constant and y the vector of other variables in 4

5 the system. We assume the system can be modeled by differential equations ẋ = f(x, y, λ) ẏ = g(x, y, λ) (2.1) where λ is a single input parameter. We also assume that (2.1) has an asymptotically stable equilibrium (x(λ), y(λ)) that depends on λ. This equilibrium exhibits homeostasis in the x-variable if x(λ) remains roughly constant as λ is varied. There is a potential singularity-theoretic approach to homeostasis because the statement the equilibrium value of some variable remains roughly constant as parameters vary near some point is roughly equivalent to the statement the derivative of the variable with respect to the parameters vanishes at that point. This leads to local quadratic or higher-order dependency; this is flatter than linear changes, and is guaranteed by the vanishing derivative. To capture this idea we say that the equilibrium (x(λ), y(λ)) of (2.1) exhibits x-homeostasis at λ 0 if x λ (λ 0) = 0 (2.2) Bifurcations, which are singularities, can be classified using singularity theory and nonlinear dynamics. Such classifications often employ changes of coordinates to put the system into some kind of normal form [2, 6, 8]. The changes of coordinates found in the literature are of several kinds, depending on what structure is being preserved. The changes of coordinates that are appropriate to the study of zeros of a map are contact equivalences [2] and the changes of coordinates that are usually appropriate to the study of dynamics of differential equations are vector field changes of coordinates [8]. A singular point of a map is a point at which the derivative vanishes, and singularity theory is the study of maps near such points. From that point of view, homeostasis looks like a singularity-theoretic phenomenon. However, there is a difficulty. In general a change of coordinates Φ(x, y) applied to (2.1) will mix the x and y variables. Although such changes of coordinates will move the equilibrium to Φ(x(λ), y(λ)), it will in general destroy the singularity condition (2.2). In order to formulate homeostasis as a singularity theory property, we must consider the following questions: (Q1) What is the appropriate class of maps (or differential equations)? (Q2) What properties should changes of coordinates preserve? (Q3) What is the characterization of such changes of coordinates? Only when these questions have been answered can we apply the theory and find out what its consequences are for homeostasis. Employing the dynamical systems or singularity-theoretic philosophy, we can ask to what extent specific examples of homeostasis are special features of specific systems and choices of variables, or whether they are in some sense invariant. Of course both 5

6 mechanisms are possible, but a systematic mathematical theory seems more appropriate when the mechanism is invariant. The starting point for this paper was the realization that in suitable circumstances x-homeostasis is an invariant of network preserving changes of coordinates. This happens because networks have distinguished variables. Specifically, we consider here questions (Q1, Q2, Q3) in the context of network dynamics. 3 Admissible Vector Fields The central concept of network theory [13, 7, 3] is that of an admissible vector field, which informally is a vector field that encodes the network topology by associating a variable with each cell and associating coupling terms between cell variables with the arrows (directed edges) of the network. The corresponding class of ODEs, which we call admissible, determine the dynamical properties of the network and this is the class that we consider in (Q1). We define networks by directed graphs where nodes stand for state spaces and arrows for coupling. Two nodes have the same node symbol if their associated state spaces are identical and two arrows have the same arrow symbol if the coupling formulae are identical. Examples of regular networks where all nodes are the same and all arrows are the same are shown in Figure Figure 2: Examples of regular networks. The admissible vector fields associated with the networks in Figure 2 are: ẋ 1 = f(x 1, x 2 ) ẋ 1 = f(x 1, x 1 ) ẋ 2 = f(x 2, x 1 ) and ẋ 2 = f(x 2, x 1 ) ẋ 3 = f(x 3, x 2 ) where x j R k and f : R k R k R k. Of course networks need not be regular. Figure 3 is an an example of a three-node network with two node types and three arrow types. The associated admissible vector fields are: ẋ 1 = f(x 1, x 2, x 3 ) x 1 R k ẋ 2 = f(x 2, x 1, x 3 ) x 2 R k (3.1) ẋ 3 = g(x 3, x 1 ) x 3 R l In previous work [13, 7, 3], we used identical nodes and arrows to classify robust synchrony, phase-shift synchrony, and bifurcations in admissible network systems. This theory was generally aimed at applications in mathematical neuroscience where identical nodes and arrows often makes sense. However, biochemical networks are special 6

7 3 1 2 Figure 3: Three-node network with two node types and three arrow types Figure 4: A four-node example with distinct node and arrow types. in this context, since node types are almost always different (they correspond to different chemical compounds), node phase spaces are one-dimensional (since nodes represent the concentration of a chemical compound), and arrow types are almost always different (they correspond to different reaction rates). An example of a four-node network in which all nodes are distinct and all arrows are different is given in Figure 4. The associated admissible vector fields are: F (x) = f 1 (x 1, x 4 ) f 2 (x 1, x 2, x 3 ) f 3 (x 1, x 2, x 3 ) f 4 (x 1, x 2, x 3, x 4 ) (3.2) In fact, the class of admissible vector fields on networks with all arrows different is easy to identify. It consists of vector fields F = (f 1,..., f n ) where f i depends explicitly on x j if and only if an arrow connects node j to node i. 4 Right Coordinate Changes and Right Cores We describe the basic concepts of the network formalism, specialized to the types of networks we consider in this paper. In [13, 7] we defined a coupled cell network to be a directed graph, whose arrows and cells may be classified into distinct types. Here we 7

8 simplify the discussion by assuming that all cells have distinct types and all arrows have distinct types. Formally: Definition 4.1. A network G is fully inhomogeneous if distinct arrows are inequivalent. Throughout this paper we assume that G is fully inhomogeneous. For simplicity we assume all cells are 1-dimensional, but the results can be extended to the general case at the expense of more complex notation. Definition 4.2. Let F : R n R n be a smooth map, and let Φ : R n R n be a diffeomorphism. The right action of Φ transforms F into G(x) = F Φ(x) Φ is right network-preserving if G is admissible whenever F is admissible. Definition 4.3. The extended input set J(i) of cell i is the set of all j such that either j = i or there exists an arrow connecting node j to node i. The extended output set O(i) is the set of all j such that either j = i or there exists an arrow connecting node i to node j. Define R(i) {j J(i) : O(j) O(i)} (4.1) Lemma 4.4. R(i) = R(i) where R(i) = J(m) (4.2) m O(i) Proof. If j R(i) then j J(m) for every m O(i), so in particular j J(i). That is, m O(j) for every m O(i), which implies O(i) O(j) and j R(i). Conversely, suppose j R(i). Then j J(i) and O(i) O(j). It follows that if m O(i), then m O(j). Or, if m O(i), then j J(m). So j R(i). Theorem 4.5. A diffeomorphism Φ = (φ 1,..., φ n ) is right network-preserving if and only if φ i (x) = φ i (x R(i) ) i (4.3) Proof. Suppose Φ is right network-preserving. We capture the restrictions on Φ = (φ 1,..., φ n ) as follows. First, let V (k) be the indices that the function φ k depends on. Fix j and let i O(j). We claim that V (j) J(i). It follows that V (j) J(i) = R(j) i O(j) Let F = [f 1,..., f n ] where f k (x) = 0 when k i and f i (x) = x j. Since j J(i), F is admissible; so G = F Φ is also admissible. Writing G = [g 1,..., g n ], it follows that 8

9 g i (x) = φ j (x) = φ j (x V (j) ) and g k (x) = 0 when k i. In order for G to be admissible we must have V (j) J(i). Conversely, let Φ be a diffeomorphism satisfying (4.3) and let F = [f 1,..., f n ] be admissible. Then let G(x) F Φ(x) = [f 1 (φ J(1) (x)),..., f n (φ J(n) (x)))] We need to show that G is admissible; that is, we need to show that g i (x) depends only on variables in J(i). We see that g i (x) = f i (φ J(i) (x)). This function can depend on a variable x k only if k R(j) for some j J(i) (or i O(j)). Since i O(j), it follows that R(j) J(i). Therefore, j J(i) and G = F Φ is admissible. Cores There is an alternative way to state the conditions of Theorem 4.5. The main point of the theorem is that a network G has a distinguished subnetwork, which determines the right network-preserving diffeomorphisms. Definition 4.6. The right core G R of a fully inhomogeneous network G is a network whose nodes are the nodes of G and whose arrows are defined as follows. An arrow j = i of G is an arrow in G R if and only if for every diagram in G of the form j = i k there exists an arrow such that j = i k In this language we can use the definition of R(i) in (4.4) to restate Theorem 4.5 as: Theorem 4.7. The right network-preserving diffeomorphisms are precisely the admissible diffeomorphisms for the right core G R. Proof. We have k O(i), and Theorem 4.5 implies that O(i) O(j). So k O(j), which means there is an arrow from j to k. Example 4.8. Consider the network in Figure 4. A short computation shows J(1) = {1, 4} O(1) = {1, 2, 3, 4} R(1) = {1} J(2) = {1, 2, 3} O(2) = {2, 3, 4} R(2) = {1, 2, 3} J(3) = {1, 2, 3} O(3) = {2, 3, 4} R(3) = {1, 2, 3} J(4) = {1, 2, 3, 4} O(4) = {1, 4} R(4) = {1, 4} The right core network for this example is shown in Figure 5. 9

10 Figure 5: The right core for the network in Figure 4. It is well known that every network can be written as a feed-forward network of transitive components. This result is equivalent to the standard relation between a preorder and a partial order, Schröder [12], wikipedia [15]. Here the preorder c d is the existence of a directed chain of arrows from cell d to cell c. The transitive components are the equivalence classes for the equivalence relation defined by c d c d and d c and the feed-forward structure is the partial order induced on equivalence classes. The right core in Figure 5 has three transitive components T 1 = {1}, T 2 = {2, 3}, and T 3 = {4}. Definition 4.9. A block in a network is a subset B of nodes such that the tail of any arrow whose head is in B is also in B. Remarks (a) If B is a block and b B then the transitive component containing b is a subset of B. So every block is a union of transitive components. Blocks have the further property that if a transitive component T B and there is some arrow from a transitive component U to a cell in T, then U B. (b) Definition 4.6 implies that blocks have a stronger property, as follows. If any two cells in the right core are joined by an arrow, then all cells in the transitive component of the tail cell of that arrow are joined to all cells in the transitive component of the head cell. All arrows point from the tail component to the head component. (c) Each block determines a subnetwork consisting of all cells in the block and all arrows whose head and tail are both in the block. (d) For example, there are three proper blocks in the right core in Figure 5: T 1, T 1 T 4, and T 1 T 2 T 3. As this example illustrates, a block may be a proper subset of another block. 10

11 Proposition Let B be a block in the right core. Let Y be the coordinates associated with the nodes in B and Z the coordinates associated with nodes not in B. Let Φ(X) be a right network-preserving diffeomorphism. Then X = (Y, Z) and Φ(X) = (Ψ(Y ), Θ(Y, Z)). Proof. Let i be a node in the block B and let φ i be the i th coordinate of the right network-preserving diffeomorphism Φ. By Theorem 4.5, φ i (X) = φ i (x R(i) ). A node j R(i) J(i) connects to i and therefore must be in B by the definition of a block. Forbidden Motifs The characterization of the right core of a network implies that unless the network G has special features, its right core G R consists solely of the cells of G with no arrows. Call this the trivial subnetwork G. Interpreted in terms of coordinate changes, it follows that for networks with trivial right core the network-preserving right-action coordinate changes are precisely the strongly admissible diffeomorphisms Φ. That is, φ c (x) = φ c (x c ) for each cell c. The condition that the right core is G is easily checked. Consider an arrow j i. This belongs to the right core if and only if O(i) O(j). That is, whenever the head cell i has an output to some cell k, the tail cell j also has an output to k. This is the content of Definition 4.6. We may state a necessary condition for the right core to be larger than G in terms of the existence of two motifs (by which we just mean simple subnetworks with a specified structure). (a) There exists a feed-forward triangle: three distinct cells i, j, k with arrows j i, i k, j k (b) There exists a bidirectional connection: two distinct cells i, j with arrows j i and i j. Case (b) arises when k = j in Definition 4.6. Theorem If a network G does not contain a triangle or bidirectional motif, then (a) The right core of G is G, that is, it comprises only its cells. (b) Every right network-preserving coordinate change is strongly admissible. Proof. The absence of the two motifs implies (a) by Definition 4.6. Clearly (a) is equivalent to (b). The absence of these two forbidden motifs is a simple sufficient condition for properties (a, b) of the theorem to occur. It is common in many networks. A necessary and sufficient condition is that there is no arrow j i with i j that satisfies O(i) O(j). Figure 6 illustrates the two motifs. 11

12 (a) (b) Figure 6: Motifs whose presence indicates a nontrivial right core. (a) The arrow j i is in the right core provided each output arrow i k from cell i has a corresponding output arrow j k from cell j. (b) Similar except that k = j. Dotted arrows indicate possible connections to and from the rest of the network. If cell i has further outputs then similar motifs must occur for j i to be in the right core. 5 System Input-Output Functions In this section we first define system input-output functions. Then we discuss the singularity theory of system input-output functions and why right equivalence of G-admissible maps is the correct equivalence to use when studying homeostasis. The results are summarized in Theorem 5.3. Suppose that Ẋ = F (X, λ) (5.1) is an admissible system of differential equations for a network depending on an input parameter λ R. In general there could be multiple input parameters in which case λ R k. Suppose that (5.1) has a stable equibrium that we assume, without loss of generality, is at the origin X = 0 = λ. The implicit function theorem implies that there exists a family of equilibria X(λ) where X(0) = 0 and F (X(λ), λ) 0. (5.2) Definition 5.1. Let B be a block in the right core, Y the coordinates associated with cells in B, and Z the coordinates associated with cells not in B. We call Y the system output. We write the implicit function X(λ) defined in (5.2) as X(λ) = (Y (λ), Z(λ)) and call Y (λ) the system input-output function a function that maps input variables λ to output variables Y. Remark 5.2. With Y -homeostasis in mind we identify any two input-output functions that differ by a constant. Consequently, in normal form, we can assume that Y (0) = 0. The system input-output function Y (λ) is defined uniquely by the family of stable equilibria and the choice of block. When B consists of a single cell, that is, Y R, we define Y -homeostasis points as points λ 0 at which dy dλ (λ 0) = 0. (5.3) 12

13 In the remainder of this section we derive the changes of coordinates that preserve the structure of system input-output functions The derivation requires several steps. The most general changes of coordinates that preserve zeros of F are contact equivalences [2]. They have the form F G(X, λ) A(X, λ)f (Φ 1 (X), Λ(λ)) λ where A(0, 0) is invertible, Λ(0) = 0, Λ (0) > 0, and Φ λ (X) Φ(X, λ) is a diffeomorphism with Φ(0, λ) 0. Since A is invertible, G = 0 reduces to H(Φ(X, λ), Λ(λ)) F (Φ 1 λ (X), Λ(λ)) = 0. Thus A does not affect the definition of the implicit function X(t) nor the definition of the system input-output map Y (λ). Define X(λ) = Φ λ (X(Λ(λ))) and compute H( X(λ), Λ(λ)) = F (Φ 1 λ ( X(λ)), Λ(λ)) = F (X(Λ(λ)), Λ(λ)) 0, where the last equality follows from (5.2). Because Y corresponds to a block in the right core and Φ is right network preserving, Proposition 4.11 implies that Φ has the form Φ(Y, Z, λ) = (Ψ(Y, λ), Θ(Y, Z, λ)). (5.4) Define Ψ λ (Y ) = Ψ(Y, λ). Then, after utilizing (5.4) and Remark 5.2, the input-output map Y (λ) transforms to Ỹ (λ) = Ψ λ (Y (Λ(λ))) + κ where κ is a constant. There is a potential difficulty. If the block B is one-dimensional, then these changes of coordinates can invalidate (5.3) unless for all λ. Specifically, the chain rule implies dỹ dλ (0) = d Ψ(Y (Λ(λ)), λ) dλ = Ψ (0, 0)dY λ=0 Y dλ (0)dΛ dλ Ψ λ (0, λ) 0 (5.5) (0) + Ψ λ Ψ (0, 0) = (0, 0), λ where the last equality follows from (5.3). Now (5.5) implies that dỹ (0) = 0 and homeostasis is preserved. We have dλ proved: Theorem 5.3. Assume the following: (a) Let B be a block of a fully inhomogeneous network G with associated coordinates Y. (b) Let X 0 be a stable equilibrium of the G-admissible system (5.1) at λ 0. 13

14 (c) Let (X(λ), λ) be the associated family of equilibria of (5.1) with X(λ 0 ) = X 0. (d) Let Y (λ) be the system input-output function associated to the block B and the family X(λ). (e) Let Φ(X, λ) Φ λ (X) be a parametrized family of network-preserving diffeomorphisms satisfying Φ(X 0, λ) = X 0 for all λ near λ 0. Let Ψ(Y, λ) Ψ λ (Y ) be the associated parametrized family of B-admissible diffeomorphisms satisfying Ψ(Y 0, λ) = Y 0 for all λ near λ 0. (f) Let Λ(λ) be a diffeomorphism satisfying Λ(λ 0 ) = λ 0. Then transforming the G-admissible map F (X, λ) to F (X) = F (Φ 1 λ (X), Λ(λ)) transforms the system input-output map Y (λ) to Ỹ (λ) = Ψ λ (Y (Λ(λ))) + κ. Moreover, infinitesimal homeostasis is preserved since dy dλ (λ 0) = 0 implies dỹ dλ (λ 0) = 0. 6 Homeostasis Revisited We have interpreted simple homeostasis as the vanishing of the derivative of the inputoutput function as an input varies. That is, 0 is a Y -homeostasis point if dy (0) = 0. It dλ is straightforward to show that we can transform the input-output function for simple homeostasis to the normal form Y (λ) = λ 2. However, examples of homeostasis suggest that a more degenerate singularity, often called a chair is also important; see Nijhout et al. [10, 11]. In the singularity setting we interpret a chair as follows. Suppose that the network coordinates are X = (Y, Z) where Y R is a block in the right core, and suppose that λ R. We call 0 a chair for Y if dy dλ (0) = d2 Y dλ (0) = 0 and d 3 Y (0) > 0 (6.1) 2 dλ3 It is straightforward to show that homeostasis points and chairs are invariant under right-left changes of coordinates since Y (λ) transforms to Ỹ (λ) = Ψ(Y (Λ)) under rightleft equivalence. We can transform a chair input-output function to the normal form Y (λ) = λ 3. Remark 6.1. Theorem 4.12 gives conditions for when the right core is the trivial network G. When this is the case, homeostasis of any cell c is a network invariant. In such networks, which are common, the most important types of homeostasis are simple homeostasis (with normal form λ 2 ) and the chair (with normal form λ 3 + aλ). 14

15 The comments of Nijhout et al. [10] suggest that there is a chair point in the body temperature data of opossums [9]. In Figure 7 note that the least squares best fit to a cubic for the brown opossum is a cubic with a maximum and a minimum, whereas the least squares fit for the eten opossum is monotone. This data suggest that in opossum space there should be a type of opossum that exhibits a chair in the system input-output function of environmental temperature to body temperature (a) (b) Figure 7: The horizontal coordinate is environmental temperature; the vertical coordinate is body temperature. (a) Data from the brown opossum; (b) data from the eten opossum. The curves are the least squares best fit of the data (from [9]) to a cubic polynomial. Informally, the codimension of a singularity is the number of conditions on derivatives that determine it. This is also the minimum number of extra variables required to specify all small perturbations of the singularity, up to suitable changes of coordinates. These perturbations can be organized into a family of maps called the universal unfolding, which has that number of extra variables. In this context simple homeostasis has codimension 0 and universal unfolding Y (λ) = λ 2. The chair has codimension 1. Its universal unfolding depends on one extra variable a and is Y a (λ) Y (λ, a) = λ 3 + aλ. (6.2) Discussion of Chair Normal Form In the singularity-theoretic setting we have chosen to define homeostasis as derivative of output with respect to input is zero at a point. In practice, homeostasis is generally interpreted as output is approximately constant over an interval of inputs. We discuss the relation between these interpretations for the chair normal form (6.2). First, for comparison, we discuss simple homeostasis. In order for the normal form λ 2 to differ from 0 by at most δ, we must have λ in the homeostasis interval ( δ 1/2, δ 1/2 ). Next, we consider how the cubic function of λ in (6.2) depends on the unfolding parameter a. The calculations are routine but the results are of interest. Figure 8 15

16 illustrates the possibilities. In this figure, δ 1 is fixed and the gray rectangle indicates the region where Y a (λ) < δ. (a) (b) (c) (d) Figure 8: Geometry of the normal form (6.2). Plot (c) corresponds to a > 0, plot (b) to a = 0, plot (a) to 0 > a > 3( δ 2 )2/3, and plot (d) to 3( δ 2 )2/3 > a. There are four plots in Figure 8. In (a,b,c) the interval on which Y a (λ) = λ 3 + aλ < δ is given by (λ 1, λ 2 ). Note that λ 2 is defined by λ aλ 2 = δ, (6.3) and since Y a (λ) is odd, λ 1 = λ 2. The transition to plot (d) in Figure 8 occurs when δ is a critical value of Y a. At that point (which occurs at a = 3( δ 2 )2/3 ) the region on which Y a is δ close to 0 splits into two intervals. We now assume ( ) 2/3 δ a > 3 (6.4) 2 Equation (6.3) implicitly defines λ 2 λ 2 (a). We claim that when a satisfies (6.4) the length of the interval on which Y a is δ-close to constant is 2λ 2 (a) and that λ 2 (a) = δ 1/3 1 3δ 1/3 a + O(a2 ). (6.5) Define the δ-homeostasis region to be the set of inputs λ such that Y a (λ) < δ 16

17 We consider only values of a for which the δ-homeostasis region is a connected interval of λ-values. We discuss how this region varies as δ tends to zero, which is one way to measure of how robust homeostasis is to small changes in parameters, or indeed to the model equations. Above we saw that for the normal form for simple homeostasis, the δ-homeostasis interval is ( δ 1/2, δ 1/2 ). We now consider this interval for a chair normal form. When a < 0 its effect is to push Y a into the region with two critical points (plot (a)), so the length of the δ-homeostasis interval increases. When a > 0 it pushes Y a into the monotone region (plot (c)) and the length of this interval decreases. It is straightforward to calculate λ 2 (0) = δ 1/3 and to use implicit differention of (6.3) to verify that dλ 2 da (0) = 1 3δ 1/3. Thus the δ-homeostasis interval for the chair is asymptotically (as δ 0) wider than that for simple homeostasis, even when a = 0. When a < 0 it becomes wider still. Remark 6.2. This suggests that a chair is a more robust form of homeostasis than simple homeostasis, which may lead to it being more common in the natural world. We suspect this fact is significant. We relate Figure 8 to the data for opossum thermoregulation. When a > 0 there are no critical points. This is in fact the case for the best-fit cubic curve to the data for the eten opossum, Figure 7(b). The slope is slightly positive at the inflection point. When a = 0, the degenerate singularity that acts as an organizing center for the unfolding, there is a unique critical point. When a < 0 there are two critical points. This is the case for the best-fit cubic curve to the data for the brown opossum, Figure 7(a). Now the slope is slightly negative at the inflection point. The transition between these two curves in opossum space is organized by the chair point a = b = 0 at which Y 0 (λ) = λ 3. In fact, data for the spiny rat give a best-fit cubic very close to this form, Figure 9. Generalized Homeostasis Suppose that a block B = {1, 2} has two cells and the input variable λ R. Then the system-input output function Y : R R 2. There are two possibilities for Ψ; namely, Ψ(y 1, y 2 ) = (ψ 1 (y 1, y 2 ), ψ 2 (y 1, y 2 )) or Ψ(y 1, y 2 ) = (ψ 1 (y 1 ), ψ 2 (y 1, y 2 )) (6.6) because of the characterization of the right core. The first case in (6.6) occurs if neither {1} or {2} is a right core block whereas the second case occurs if {1} is also a block for 17

18 Figure 9: Thermoregulation in the spiny rat. The horizontal coordinate is environmental temperature; the vertical coordinate is body temperature. The curve is the least squares best fit of the data (from [9]) to a cubic polynomial. the right core, as happens for the {1, 4} block in Figure 4. In the latter case there is a codimension one homeostasis-like possibility where dy 1 dλ (0) = 0 = dy 2 dλ (0) simultaneously. The first case reduces to studying plane curves (Y (λ) = (Y 1 (λ), Y 2 (λ))) under right-left equivalence. It will be interesting to see whether and block structures (other than the relatively straightforward one of a singleton block) leads to network invariants that are of biological interest. 7 Computation of Homeostasis Points Suppose that F is a network admissible vector field and let (X 0, λ 0 ) be a stable equilibrium. Suppose that B is a one-dimensional block with coordinate Y and Z is the vector of node coordinates not in B. Let X 0 = (Y 0, Z 0 ) and let X(λ) = (Y (λ), Z(λ)) be the λ family of equilibria satisfying (Y (λ 0 ), Z(λ 0 )) = X 0. Then, in general, the input-output function Y (λ) has a Y -chair at λ 0 if dy dλ (λ 0) = d2 Y dλ 2 (λ 0) = 0 and d 3 Y dλ 3 (λ 0) > 0 (7.1) The derivatives in (7.1) can be computed from the vector field F by using implicit differentiation just as one does in bifurcation theory. Let A = (DF )(Y 0, Z 0, λ 0 ). Then differentiating F (Y (λ), Z(λ), λ) = 0 with respect to λ yields ] [ Yλ Z λ = A 1 F λ 18

19 Similarly, the second implicit differentiation at a point where Y λ = 0 yields [ ] ( ([ ] [ ]) [ ] ) Yλλ = A 1 (D 2 F ), + (DF λ ) + F λλ Z λλ Z λ Z λ Z λ evaluated at (Y 0, Z 0, λ 0 ). writing out the formula. The third derivative is generally nonzero; we won t bother 8 A Simple Example Consider a made up model (Figure 10(a)) of thermoregulation with state variables Y, Z 1, Z 2, of which Y is the output variable (body temperature) and Z 1, Z 2 are other (regulatory) variables. This example shows that we can have homeostasis in one variable but not in the others. Let λ be the input parameter (external temperature) and γ an internal parameter. Suppose the dynamic equations are: Ẏ = Y Z 2 Z 1+γ 1+Z 2 1 Ż 1 = Z 1 + Z 2 2λ Ż 2 = Z 1 + Z 2 (8.1) Z Z 1 2 Z Z 1 2 Y Y (a) (b) Figure 10: A simple made up example of three cells As a network, this has three cells. Cells Z 1, Z 2 are connected bidirectionally and both feed forward to cell Y. The right core (Figure 10(b)) has two blocks: {Z 1, Z 2 } is a full 2 2 block, and {Y } is a 1 1 block. So homeostasis on Y makes sense. We claim that there is a point of homeostasis in (8.1) at γ = λ 0 = 0. At equilibrium the second and third equations give Substitute in the first equation to get Z 1 = Z 2 = λ. Y = λ + λ + γ 1 + λ 2 Y γ(λ) 19

20 We show that this equation has a chair. The conditions for a chair are: 0 = dy = 1 + λ2 +2γλ 1 dλ (λ 2 +1) 2 0 = d2 Y = 2λ3 +6γλ 2 6λ 2γ (8.2) dλ 2 (λ 2 +1) 3 Note that (8.2) has a (unique) solution at λ = γ = 0. Thus Y γ (λ) has a chair point or something more degenerate at λ = γ = 0. Since d 3 Y/dλ 3 = 6 > 0 at γ = λ = 0, Y 0 (λ) has a chair point at λ = 0. Note that at equilibrium dz 1 /dλ = dz 2 /dλ = 1 for all λ and γ; so homeostasis in one cell variable need not imply homeostasis in other cell variables. 9 Remarks and Problems The ideas and methods of this paper generalize to give a variety of network-invariant phenomena, in the sense that they are preserved by network-invariant coordinate changes. Here we focused on simple homeostasis and chair points, which are the lowest-codimension singularities when there is one input variable λ and one output variable Y. Here we required the Y -cell of the network to be in a block of size 1 in order to justify the coordinate changes induced on the input-output function. This approach can be generalized in at least three ways: Consider more degenerate singularities. Consider multiple input variables λ = (λ 1,..., λ p ). Consider multiple output variables Y = (Y 1,..., Y q ) in the same cell; that is, allow the cell phase space to be R q where q > 1. Consider more than one cell simultaneously. The cells concerned should form a block to obtain sensible coordinate changes. All of these generalizations have their own special features. However, they all fall in to the same general mathematical setting: singularities of maps Y : R p R q near the origin of R p. The coordinate changes that emerge from the previous discussion are a form of contact equivalence on input-output functions: Ỹ (λ) = Ψ λ (Y (Λ(λ))) In the case of a block of size 1 and a single input variable, the singularity classification reduces to that of maps R R, which are straightforward generalizations of the chair. The normal forms are ±λ k, which occur when the first k 1 derivatives of Y vanish and 20

21 the kth is positive or negative but not zero. The codimension is k 2 and the universal unfolding is λ k + a k 2 λ k a 1 λ The even k cases are analogous to simple homeostasis, the odd k cases are analogous to chairs. Larger values of k give flatter behavior near the singularity. When the block concerned is more complicated that the size 1 block considered in this paper, the transformations Ψ λ are restricted by the form of the right core of the original network. In the simplest case, consider two cells Y 1, Y 2 that form a block. There are three distinct possibilities: (a) Each of {Y 1 }, {Y 2 } is a block. Here we in effect are studying two independent blocks of size 1 but with the same input variables. In the right core there are no connections between Y 1 and Y 2. The transformations Ψ λ take the form Ψ λ (Y 1, Y 2 ) = (Ψ 1 λ(y 1 ), Ψ 2 λ(y 2 )) (b) Each of {Y 1 }, {Y 1, Y 2 } are blocks the triangular case, in which the right core leads to a feed-forward network with Y 1 connecting to Y 2. Now Ψ λ takes the form Ψ λ (Y 1, Y 2 ) = (Ψ 1 λ(y 1 ), Ψ 2 λ(y 1, Y 2 )) (c) {Y 1, Y 2 } is a block, but neither of {Y 1 }, {Y 2 } is a block. In the right core there are connections both ways between Y 1, Y 2. Now Ψ λ takes the form Ψ λ (Y 1, Y 2 ) = (Ψ 1 λ(y 1, Y 2 ), Ψ 2 λ(y 1, Y 2 )) With more than two blocks the possibilities become more complex. For each such setting, there is a singularity-theoretic classification of the possible local structures for the input-output map, organized by increasing codimension. Each has a universal unfolding whose geometry determines the invariant phenomena. These issues require singularity-theoretic techniques and we plan to investigate them in [5]. The resulting normal forms include certain types of homeostasis, in which (for example) several inputs keep several outputs approximately constant simultaneously. However, the possible singularity types are not limited to such behaviour. At the present time the implications for experiments and observations are not clear. Acknowledgements We thank Mike Reed and Janet Best for many helpful conversations in particular for an introduction to the notion of a chair. This research was supported in part by the National Science Foundation Grant DMS to the Mathematical Biosciences Institute. 21

22 References [1] J. Best, H.F. Nijhout, and M. Reed. Homeostatic mechanisms in dopamine synthesis and release: a mathematical model, Theor. Biol. Med. Model. 6 (2009) 20pp. [2] M. Golubitsky and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory I, Applied Mathematics Series 51, Springer, New York [3] M. Golubitsky and I. Stewart. Nonlinear dynamics of networks: the groupoid formalism, Bull. Amer. Math. Soc. 43 (2006) [4] M. Golubitsky and I. Stewart. Coordinate changes for network dynamics, preprint [5] M. Golubitsky and I. Stewart. The singularity theory of homeostasis, preprint [6] M. Golubitsky, I. Stewart, and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory II, Applied Mathematics Series, 69, Springer, New York [7] M. Golubitsky, I. Stewart, and A. Török. Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dynam. Sys. 4 (2005) [8] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York [9] P.R. Morrison. Temperature regulation in three Central American mammals. J. Cell Compar. Physiol. 27 (1946) [10] H.F. Nijhout, J. Best, and M.C. Reed. Escape from homeostasis. Math. Biosci. 257 (2014) [11] H.F. Nijhout, M.C. Reed, P. Budu, and C. Ulrich. A mathematical model of the Folate cycle New insights into Folate homeostasis, J. Biological Chemistry 279 (2004) [12] B.S.W. Schröder. Ordered Sets: An Introduction, Birkhäuser, Boston [13] I. Stewart, M. Golubitsky, and M. Pivato. Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dynam. Sys. 2 (2003) [14] physiological homeostasis.htm (updated 2000). [15] en.wikipedia.org/wiki/preorder 22