Control theory meets connectomes?

My colleagues and I have been working through this intriguing paper [1] from a few weeks ago:

Yan, G., Vértes, P.E., Towlson, E.K., Chew, Y.L., Walker, D.S., Schafer, W.R., and Barabási, A.-L. (2017). Network control principles predict neuron function in the Caenorhabditis elegans connectome. Nature advance online publication.

This seems like a very important contribution. It promises detailed insights about the function of a neural circuit based on its connectome alone, without knowing any of the synaptic strengths. The predictions extend to the role that individual neurons play for the circuit’s operation. Seeing how a great deal of effort is now going into acquiring connectomes [2] – mostly lacking annotations of synaptic strengths – this approach could be very powerful.

The starting point is Barabási’s “structural controllability theory” [3], which makes statements about the control of linear networks. Roughly speaking a network is controllable if its output nodes can be driven into any desired state by manipulating the input nodes. Obviously controllability depends on the entire set of connections from inputs to outputs. Structural controllability theory derives some conclusions from knowing only which connections have non-zero weight. This seems like a match made in heaven for the structural connectomes of neural circuits derived from electron microscopic reconstructions. In these data sets one can tell which neurons are connected but not what the strength is of those connections, or even whether they are excitatory or inhibitory. Unfortunately the match is looking more like a forced marriage…

1. Linearization around a fixed worm: An early step in the modeling of the worm is to linearize the dynamic equations around a fixed point (Eqn 2), where all the variables are stationary. Why a fixed point? All the measurements here are taken on swimming worms. In the phase space of muscle activations the worm executes a closed trajectory, rather than sitting at a fixed point. It seems more appropriate to analyze perturbations about this closed trajectory.

Is linearization even justified at all? The swimming worm operates in a fully nonlinear regime, with the motorneurons depolarized and hyperpolarized all the way. As the authors say, a linearly controllable system is also nonlinearly controllable. But the logic does not work in the other direction [4], so conclusions about non-controllability from a linear approximation cannot be carried over to the nonlinear system. This, however, is what the article does all the time.

2. All neurons are perfect integrators: The next set of difficulties emerges in the article’s tutorial material. In Figure 2a we are shown one input neuron that drives two output neurons, and we are told that the output neurons are not controllable. This seems odd. Suppose for example that one of the output neurons responds better to low frequencies and the other to high frequencies, then one can design an input signal that drives each output independently of the other.

Digging into the figure caption we find that the derivatives of the 2 output neurons are strictly proportional to each other, which doesn’t seem right either. What this relation implies is that the neurons have no internal dynamics. The diagonal matrix elements a₂₂ and a₃₃ in Eqn (2) are assumed to be zero. In other words the neurons must be perfect integrators with infinite time constant. That doesn’t describe any real neural system. Approximately every model of a point neuron (i.e. with just one dynamical variable) starts with

where  is the neuron’s integration time, and  is the synaptic current produced by inputs from other neurons [5, Ch 7]. Typical integration times range from milliseconds to hundreds of milliseconds, but they are not infinite. In fact, an entire subfield of neuroscience is contemplating how brain circuits can sustain stable activity patterns with neurons that only have short-term dynamics.

Digging into the paper’s supplement we find that the theory could be rescued if is finite but identical for every neuron in the circuit. Again, this is a patently unrealistic assumption: Neural networks have diverse components with many time constants even within the same circuit [6], and especially circuits for pattern generation benefit from elements with different dynamics. For example, a computational model for the C elegans swimming circuit [7] uses point neurons with time constants that range over a factor of 3.5. So the condition imposed here that all the time constants be identical just doesn’t apply to any real neural systems, or as the authors might say (see Suppl II.A) only to “some pathological cases for which the algebraic variety in parameter space has Lebesgue measure zero”.

Note that this problem with “structural controllability theory”, namely the insistence on nodes without internal dynamics, was spotted shortly after Barabási’s 2011 paper [3], and the resulting claims have already been refuted in the literature [8]. It was pointed out that in a system where the nodes are allowed to have dynamics a single time-dependent input is sufficient to make the network “structurally controllable”. Why does that same theory get dished up again, with all the same deficiencies?

3. Experimental measurements incommensurate with theory: Suppose we overlooked this fatal flaw and got to the point of considering experimental data against the theory’s predictions. The theory’s output is categorical: either a set of nodes is controllable or it is not. That is because the theory’s inputs are categorical: either two nodes are connected or not. Because no quantitative information is available about the strength of synapses in the network, we can expect no quantitative predictions about the network’s dynamics. So we must look for experimental proof that the worm has gone from controllable under one condition to uncontrollable under another. Nothing of the sort happens in the reported experiments.

The worms swim just fine both before and after the network perturbations that are supposed to cause uncontrollability. There are small changes in the shape of the swimming worm (Figs 2f and 3b-c, example above): All the systematic effects of neuron ablation are smaller than the natural variation within a set of normal worms. So if the normal worms (red dots) are considered controllable, the perturbed worms (green dots) are controllable too. There is no logical connection between a small quantitative change in the worm’s swimming movements and the all-or-nothing predictions from structural controllability theory.

One might ask whether measurements of the type reported here can serve to test controllability even in principle? In the intact worm 89 muscles are independently controllable, so their activations could fill an 89-dimensional volume. But the measurements cover only a 4-dimensional subspace spanned by the “eigenworms”. How can one assess controllability of the system by observing only a tiny fraction of its degrees of freedom? Suppose a perturbation of the network leads to loss of one of those 89 dimensions. For that effect to be detected, the lost dimension must reside inside the 4-dimensional subspace of the measurements. Again that happens only in “some pathological cases for which the algebraic variety in parameter space has Lebesgue measure zero”.

Summary: It seems that “structural controllability theory” doesn’t apply to real neural systems, and that the experimental measurements reported here have no logical relation to the predictions of the theory. So this paper leaves plenty of room for discoveries, and the problem of linking connectomes to functional predictions remains an open challenge.

[1] Yan, G., Vértes, P.E., Towlson, E.K., Chew, Y.L., Walker, D.S., Schafer, W.R., and Barabási, A.-L. (2017). Network control principles predict neuron function in the Caenorhabditis elegans connectome. Nature advance online publication.

[2] Swanson, L.W., and Lichtman, J.W. (2016). From Cajal to Connectome and Beyond. Annual Review of Neuroscience 39, 197–216.

[3] Liu, Y.-Y., Slotine, J.-J., and Barabási, A.-L. (2011). Controllability of complex networks. Nature 473, 167.

[4] Coron, J.-M. (2007). Control and Nonlinearity (American Mathematical Society).

[5] Dayan, P., and Abbott, L.F. (2001). Theoretical neuroscience: computational and mathematical modeling of neural systems (Cambridge, Mass.: MIT Press).

[6] Neuroelectro.org

[7] Karbowski, J., Schindelman, G., Cronin, C.J., Seah, A., and Sternberg, P.W. (2008). Systems level circuit model of C. elegans undulatory locomotion: mathematical modeling and molecular genetics. J Comput Neurosci 24, 253–276.

[8] Cowan, N.J., Chastain, E.J., Vilhena, D.A., Freudenberg, J.S., and Bergstrom, C.T. (2012). Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks. PLOS ONE 7, e38398.