Homeostasis occurs when an observable of a system (such as inner body temperature) remains approximately constant over a range of an external parameter (such as ambient temperature). More precisely, homeostasis refers to a phenomenon whereby the output x_o of a system is approximately constant on variation of an input I. Homeostatic phenomena are ubiquitous in biochemical networks of differential equations and these networks can be abstracted as digraphs G with a fixed input node \iota and a different fixed output node o . We assume that only the input node depends explicitly on \mathcal{I x_o(I) . We then study infinitesimal homeostasis : points \mathcal{I_0 where dx_o{d\mathcal{I}}(\mathcal{I}} _0. Stability implies that there is a stable equilibrium X(\mathcal{I}) for each \mathcal{I} near \mathcal{I}_0 and infinitesimal homeostasis occurs at \mathcal{I}_0 when (dx_o/d\mathcal{I})(\mathcal{I}} _0) = 0 by showing that there is a square homeostasis matrix H associated to \mathcal{G) for which dx_o/d\mathcal{I} = 0 if and only if } \det(H) = 0. Applying combinatorial matrix theory and graph theory to H allows us to classify types of homeostasis. We prove that the homeostasis types correspond to a set of irreducible blocks in H each associated with a subnetwork and these subnetworks. Specifically, we prove that each factor corresponds to a subnetwork of \mathcal{G}. The factors } divide into two  classes: structural and appendage. For example, a feedforward loop motif is a structural type whereas a negative feedback loop motif is an appendage type. We give two algorithms for determining a menu of homeostasis types that are possible in \mathcal{G: one algorithm enumerates the structural types and one enumerates the appendage types. These subnetworks can be read directly from \mathcal{G} without performing calculations }  on model equations  .
Homeostasis refers to a phenomenon whereby the output x_o of a system is approximately constant on variation of an input I. 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 G with a distinguished input node \iota , a different distinguished output node o , and a number of regulatory nodes \rho_1,\ldots,\rho_n. In these models the input-output map x_o(I) _0 where dx_o{d\mathcal{I}}(\mathcal{I}} is defined by a stable equilibrium X_0 at \mathcal{I_0. Stability implies that there is a stable equilibrium X(\mathcal{I}) for each \mathcal{I} near \mathcal{I}_0 and infinitesimal homeostasis occurs at \mathcal{I}_0 when (dx_o/d\mathcal{I})(\mathcal{I}} _0) = 0. We show that there is an (n+1)\times(n+1) homeostasis matrix H (\mathcal{I) for which dx_o/d\mathcal{I} = 0 if and only if } \det(H) = 0. We use combinatorial matrix theory to factor the polynomial \det(H) and thereby determine a menu of different types of possible homeostasis associated with each digraph \mathcal{G. Specifically, we prove that each factor corresponds to a subnetwork of \mathcal{G}. The factors } divide into two combinatorially defined classes: structural and appendage. : one algorithm enumerates the structural types and one enumerates the appendage types. These subnetworks can be read directly from \mathcal{G} without performing calculations } Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostatic subnetwork motif corresponding to each factor of \det(H) without performing numerical simulations on model equations . The algorithm allows us to classify low degree factors. 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) .
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