The Geometry of Nervous Systems may have Precisely Evolved to Support Rich Dynamic Function
There are many methods and tools available to study complex systems. But understanding how the structural connectivity of a network constrains the dynamics the network is able to support is a difficult question to formulate and answer, and remains a very active and open area of research.
In a recent paper published in Frontiers in Systems Neuroscience we introduced a novel set of quantitative analyses methods to address this. And we applied these tools to an analysis of the dynamics of the Caenorhabditis elegans worm connectome.
The C. elegans connectome — the connectivity map of the network of neurons that make up its nervous system — consists of 302 neurons and their anatomical links. Although C. elegans’ connectome has been known for decades, how neurons functionally interact in the context of the entire network and how the resultant dynamics regulates function is not fully understood. A recent trend in C. elegans research is to study how the firing dynamics of its neurons interact in context with the known connectivity of the connectome, rather than drawing inferences from just the structural connectivity of the network itself.
In this work, we specifically attempted to answer the following question: How does the concurrent activity of independent neuronal elements (nodes in the network), along axons (the edges or ‘links’ in the network) ultimately give rise to a rich behavioral repertoire?
We took advantage of the dynamic implications — in other words, the necessary resultant dynamics — of the spatial separation of the neurons that make up C. elegans nervous system. Although the physiology C. elegans of neurons is complicated and at times even controversial among experts (they do not fire action potentials in a classical way), we intentionally took a naive and simple route and assumed that communication between neurons is constrained by their axonal conduction velocity. The speed of the signals between neurons, coupled with the spatial separation between neurons, results in edge signal delays, i.e. signaling latencies that limit how fast one neuron can communicate with another neuron it is connected to. We modeled the signaling dynamics in the network as the passing of a discrete quantal unit, at finite speeds, between neurons, along directed edges. As such, node interactions were constrained by the network’s connectivity, spatial geometry, and signaling parameters.
This analysis is built on a theoretical model and framework developed by our group that, among other things, is able to describe and compute how local interactions among connected nodes give rise to global dynamics in an emergent way. This framework was derived from canonical principles of spatial and temporal summation in biological neurons. The network interactions are governed by the arrival times of incident signals into nodes and how those signals compete to activate downstream nodes. We used the model in this work to compute the network dynamics on the C. elegans connectome.
We constructed a geometric network by combining the C. elegans wiring diagram–focusing on axonal connectivity– with node location data to calculate Euclidean straight-line edge lengths. We then used the edge lengths and signal conduction velocities to calculate edge signal delays. We assumed every neuron (node) had a same refractory period — a period of time during which it cannot respond to incoming signals. The refractory period was taken from a realistic biological range. Finally, each node responded in either an excitatory or inhibitory manner.
Using this geometric network model, we studied the dynamics that resulted from stimulating a chemosensory neuron called ASEL. This neuron is known to detect the presence of ‘food’. Laboratory experiments have shown that the activation of ASEL ultimately results in a behavioral motor response: C. elegans moves towards the food source. This locomotion is produced by the synchronized activation of mid-body motor neurons.
We found that a geometric embedding of the network, in contrast to considering just its connectivity structure, increases the dynamical repertoire the network is able to produce. We did this by calculating the number of unique network states. We quantified the number of network states between a spatially-aware network — one whose edge delays are derived from node location data — and a spatially-unaware network, one whose edge delays are all the same. Interestingly, stimulating ASEL led to dynamics on the spatially-aware network that eventually resulted in contralateral motor neuron activation in the ventral (VB) and dorsal (DB) classes of motor neurons. This rhythmic alternating back-and- forth motor neuron firing pattern is qualitatively similar to that observed experimentally that is necessary for movement of the worm (towards a food source in this case).
The Take Home Message
This result is subtle but critical in its interpretation. (Or at least in how we interpreted it.) It is important to understand that we did not in any way model or intentionally try to emulate the rhythmic oscillatory dynamics between the two motor neuron populations that behaviorally produce the locomotion that moves C. elegans towards the food source. This dynamic behavior was in response to a single impulse input into the network (via ASEL), and reflects the inherent (and we propose) purposeful structural wiring of the C. elegans connectome that has evolved over a very long time to serve purposeful behavioral functions. In other words, the connectome of C. elegans, being far from random, somehow evolved to be just right to support the perfect dynamics that allows it to survive and thrive in its environment. It is fascinating to think how this might have happened. And how a similar effect might be the case in the nervous systems and brains of other organisms, including ours.
How We Did This Analysis
To uncover how the C. elegans connectome and dynamics give rise such inherent and emergent contralateral motor activation patterns, we identified and analyzed the signaling paths beginning at ASEL to the VB and DB neurons in a new way.
Neuronal signaling paths encode the causal chain of node activations over the course of the dynamics. To describe each signaling path we introduced the notion of a Temporal Sequence (TSeq). Each TSeq is a temporally ordered sequence of nodes, formally a walk on a graph. We developed a method to quantitatively compare TSeqs, which we used to determine the extent to which subnetworks preserve causal signaling paths present in the connectome. Furthermore, we decomposed the complex network activity into a compact basis set of TSeqs. This set plays two roles, it can be used as a signature for the network dynamics, and it can be used to construct subnetworks which more closely resemble the signaling dynamics of the larger network.
This piece is part of the collection ‘The Technical Paper Reboot’ — Short adaptations of some of our technical papers that in particular highlight the limits and boundaries of neuroscience. We refer the reader to the technical papers in the links in the article for full details and citations.
