Many aspects of animal physiology involve the longitudinal propagation of rhythmic time-periodic patterns in which linear chains of neurons oscillate in synchrony or with specific phase relations. These two types of behaviour can be interpreted as standing waves and traveling waves, respectively. A common mechanism for such propagating chains involves a network of neurons, often called a central pattern generator (CPG), which generates the basic rhythms such as, for example, coordinated leg movements in insect locomotion. This lies at the start of a feedforward network along which the CPG signals propagate.
The main aim of this work is to describe a general method for constructing networks in which periodic dynamics of a specified CPG propagates synchronously, or phase-synchronously with a regular pattern of phase shifts, along a feedforward chain, tree, or any other feedforward structure. (For simplicity we often use the term “chain” without implying linear topology.) This is achieved by constructing the rest of the network as a feedforward lift of the CPG.
Of course, the use of chains propogate signals is not a new idea, as even a cursory glance of the literature shows. Indeed, it is arguably the simpest, most natural, and most obvious method. However, the formal setting in which the analysis is carried out in this work makes it possible to prove some general stability results and helps to unify the area.
We can consider the toy network below as a working example, where the CPG is represented by three coupled cells (cells 1, 2 and 3), and the feedforward chain comes off from it (cells 4, 5, 6 and 7). In the diagram below there is only one node type and one arrow type and the colours show a synchrony pattern (a “balanced colouring” where, for example, cell 6 can be associated with cell 3, they both are fed by the same cell/arrow combination where that cell is coloured grey).
If each cell is identical then the system of equations describing the dynamics of it might be written:
\[\begin{equation}
\begin{array}{rcl}
\dot{x}_1 &=& f(x_1,x_3) \\
\dot{x}_2 &=& f(x_2,x_1) \\
\dot{x}_3 &=& f(x_3,x_2) \\
\dot{x}_4 &=& f(x_4,x_3) \\
\dot{x}_5 &=& f(x_5,x_4) \\
\dot{x}_6 &=& f(x_6,x_5) \\
\dot{x}_7 &=& f(x_7,x_6)
\end{array}
\end{equation}\]
A key part of such networks are conditions under which we ensure that these patterns are fed forward in a stable manner, it turns out that this happens quite naturally when considering different notions of stability for equilibia and periodic orbits. The main technical difficulties arise when considering transverse stabilties (perturbations transverse to the synchrony subspace). An important feature of feedforward lifts is that tranverse Floquet stability is determined by the dynamics of individual nodes of the CPG. In consequence, results show that if the propogating signal is Floquet stable along one step of the chain, then it remains Floquet stable however long the chain is, or if the chain branches like a tree.
For the toy model above the chain can be reduced to the following quotient network for the “balanced colouring” given in the previous figure.
We expect solutions to such a network to consist of identical oscillations each a third a period out of phase with the previous which is the pattern that will this propogate further along the chain.
References:
- I. Stewart, D. Wood, Stable Synchronous Propagation of Signals by Feedforward Networks, SIAM Journal on Applied Dynamical Systems 23, pp167-204, 2024, https://doi.org/10.1137/23M1552267.
- I. Stewart and D. Wood, Stable Synchronous Propagation of Signals in Feedforward Networks of Standard Model Neurons. In preparation
