Continuous-time recurrent ‘neural’ network as a model of intracellular signalling

This is a bit of a long post (took me the better part of today). It shall serve as my initial train of thought on something that I would like to refine, extend, and publish somewhere more seriously. So, if you find the post interesting, and/or you think you can help me shape it such that it becomes so, then don’t hesitate to contact me.

In Beer’s 1995 paper On the dynamics of small continuous-time recurrent neural networks, it was suggested that small networks could be used as “powerful building blocks for the modular construction of larger networks with desired dynamics” (p. 29).

To my knowledge, this suggestion has not been followed up. I suspect one reason for this, is that although useful for designer or engineer; the suggestion appears to be rather ad hoc or perhaps removed from any biological grounding. In this post, I provide one possible biological interpretation for its use with some support from a set of concrete experiments.

Bacterial chemotaxis, adaptation, and E. coli’s protein circuit

Bacteria are attracted (or repelled) by certain chemicals. When a pipette with nutrients is placed in a plate of swimming E. coli, they cluster around its mouth. If the chemical is noxious, they swim away from it. Bacterial chemotaxis has been studied in-depth for well over four decades.

Bacteria move by swimming relatively straight, in what are called ‘runs’. They can also turn in a random new direction. This behaviour is called ‘tumbling’. Because bacteria are too small to sense the gradient along their own body, they use the temporal gradients to guide their motion. The strategy that they use to do chemotaxis is to reduce or increase their tumbling frequency, depending on whether the concentration of the chemical has increased or decreased, respectively.

The tumbling frequency is independent of the absolute levels of attractant concentration. In other words, after an initial decrease, when attractant is added to the environment, the tumbling frequency returns to the same level as before attractant was added. This is referred to as ‘exact adaptation’.

The chemotaxis signal transduction network works as follows (see figure below). The input to the circuit is the attractant concentration. The output is the tumbling frequency. The receptors (which sense the input) bind to a protein kinase called CheA. Binding of the attractant makes CheA inactive. CheA can transit rapidly between active and inactive states. When CheA is active, it phosphorylates another protein called CheY. Phosphorylated CheY increases the tumbling frequency. Another protein, CheZ, removes the phosphorylation of CheY.

E. coli's chemotaxis protein circuit taken from Alon (1999, p. 139). Reproduced without permission.

The response time from input, through CheA and CheY, to tumbling frequency occurs within 0.1 seconds. This part of the network accounts for an important part of bacterial chemotaxis: attractants lead to a reduction in tumbling frequency. But it does not yet explain its ability to adapt to different absolute concentrations.

In order to understand adaptation, we will need to consider a second  – much slower – pathway. For this we need to consider two other proteins: CheR, and CheB. As we just saw, increased attractant concentration can leave the receptors and CheA inactive, and thus irresponsive to a further increase in concentration. But the receptors can become active again by methylation modifications. Protein CheR can methylate the receptors, while enzyme CheB can remove the methylation. CheR concentration is relatively constant. So methyl groups are added at some unchanged rate. However, CheA affects CheB. Active CheA phosporilates CheB. This means that when CheA becomes inactive, phosporilation of CheB is reduced, and the receptors are allowed to become methylated. This happens at a much slower rate, in the orders of minutes. The increase of methylated receptors increases the tumbling frequency, slowly returning it to its steady-state. This is effectively a negative feedback loop from CheA to CheB.

So there are three key players: CheA (and receptors), CheY (and the flagella motors), and CheB. And two key pathways: a fast and a slow one.

This has been only a very rough sketch. For a good introduction and further details read Chapter 7: “Robustness of protein circuits” in Uri Alon’s An introduction to systems biology.

An artificially evolved CTRNN version and its relation to the biochemical mechanisms in E. coli

Over the last two decades, a range of computer models of the chemotaxis signal-transduction pathway in E. coli have been published. Dennis Bray summarises the most important ones here. Perhaps surprisingly, none model the conditions (i.e. environmental situations, fitness functions) under which the adaptation mechanism can evolve (at least not that I’m aware of).

I will consider a model body and environment very similar to that used in most of the bacterial chemotaxis studies done so far. But instead of using artificial chemistries or kinetic equations (e.g. Michaelis-Menten) to model their internal dynamics, we will use continuous-time recurrent nonlinear networks (Beer 1995):

CTRNN equation
CTRNN equation

An agent is modelled as a point in a 2D circular environment (sort of like a Petri dish, but instead of walls it wraps around in diametrically opposed points). The activity of one of the network’s component determines the probability that it will continue to swim straight (with some noise) or tumble (i.e. change to a new random direction). The activity of another one of the components will serve as the receptor, sensing the chemical concentration in the surroundings of the agent.

If the absolute levels of the chemical concentration are kept fixed throughout evolution, then a ‘reactive’ strategy (i.e. not requiring adaptation) can be successful. For example, if the peak of chemical concentration is always 1, then one successful strategy would be swim straight most of the time, then increase the tumbling frequency to the maximum (so as to stay near that area) when the chemical concentration reaches close to 1.

If the absolute level is allowed to vary, so that sometimes the peak is at 1, but in another Petri dish it can be higher. Then knowing that you are in chemical concentration 1, is not enough to know whether you are in the peak. You have to keep following the gradient up all the time.

The range of different types of environments that the bacteria can experience during evolution.
The range of different types of environments that the bacteria can experience during evolution.

For that reason, the chemical concentration can vary during evolution. The variation included in the experiments described ahead is minimal (i.e. the lowest point in the gradient can range between [0, 1], while the peak can range between [1, 2]. The steepness remains the same at all times, see figure below).  E. coli can adapt to around 4 or 5 orders of magnitude of absolute levels of concentration. But all that is relevant for our purposes is that there is some variation, as an adaptation mechanism will be required as long as there is some variation.

Artificially evolving successful 3-component networks on this task was not hard. Most interestingly, the most successful ones all evolved some resemblance with E. coli’s protein circuit. See the architecture for the best evolved network in the figure below. Each component is represented by a circle. The size of the circle represents its time-constant. Two of the components (the sensor and motor ones) are always as fast as allowed. The extra component always evolved to be at least two orders of magnitude slower. The connections between the components can be interpreted as one component promoting the production of another, in the case of the black arrows. And depression or inhibition in the case of the gray ones. Self-promotion/inhibition links are not shown. Also, the thickness of the lines in the links represents the strength of the promotion/depression.

Architecture of the best evolved network. See the paragraph above for an explanation.
Architecture of the best evolved network. See the paragraph above for an explanation.

We can also constrain the network to have any of all possible ways to connect the three nodes and observe their ability to evolve and perform well at the task. Although I won’t go into any detail about that here, the two important components that make the difference are: (a) the link between the sensory input and the motor output, and (b) the negative feedback loop through an extra component that is usually around 2 orders of magnitude slower than the rest.

In the movie below, you can observe the artificial bacteria in action, as well as the activity of its internal components. We show ten trials without resetting the state of the circuit. After each trial, the agent is dropped in a random new position. In the movie, this happens every 20 seconds. Also, at each trial the absolute level of the chemical gradient is increased.

On the left, we have the simulated ‘Petri dish’. The different shades of black represent the chemical concentration: darker regions have more. The black circle represents the end of the dish. The blue circle represents the point of highest concentration (where the mouth of the pipette would be). The orange disk represents the simulated bacteria. On the right, the activity of each of the components can be observed. The coloured circles correspond to the maximum level of activity or concentration that a component can reach. The colours are the same used for the architecture figure above. The black disk that varies in size inside each circle represents the level of activity or concentration.

In order to understand how the evolved circuit works, it will be useful to observe its activity during the following simple experiment. The bacterium is placed in an environment without a gradient (concentration is uniform). After some time, each of the components settle to their steady-state.

We add an attractant (such as aspartate) to the liquid uniformly in space. The attractant concentration increases from 0 to 0.5, but no spatial gradients are formed. The cell senses an increase in attractant levels, no matter which direction they are swimming. It ‘thinks’ things are getting better and suppresses tumbling. After a while, the cell realizes it has been fooled. The tumbling frequency is increased again, even though attractant is still present at the same level.

The activity of the network over time as the concentration is increased in steps. The colours correspond to the previous architecture figure.
The activity of the network over time as the concentration is increased in steps. The colours correspond to the previous architecture figure.

After we have let the cell settle down again, and the component that controls the tumbling frequency (blue trajectory) returns to a similar value as before the step, we add again more attractant. And so on.

We can perform the same experiment but with down steps instead. Instead of decreases in tumbling frequency, what we observe are increases in tumbling frequency, as expected.

The activity of the network over time as the concentration is decreased in steps.
The activity of the network over time as the concentration is decreased in steps.

This is the process that is called adaptation (see Alon, 2007). Bacterial chemotaxis shows it: the tumbling frequency in the presence of attractant returns to the same level as before attractant was added. In other words, the steady-state tumbling frequency is independent of attractant levels. Except for the border cases, this is true in the evolved model.

Notice that in our evolved circuit near-perfect adaptation is observable at most of the absolute levels of concentration, except for the limits that is experienced during evolution. We would think this is expected in E. coli as well, although I believe this has not been studied. Such a thing would require understanding (and experimenting with) the ecological niche that E. coli has been exposed to during evolution.

An important point of the model is the following. There are analogous mechanisms operating in this evolved model in relation to what is understood in E. coli’s protein circuit. The sensory node (in red) could be said to implement the concentration of inactive CheA. In the sense that, attractant molecules bind to CheA making more of inactive. The output node (in blue) could be said to implement the concentration of phosporilated CheY. The more there is, the higher the tumbling frequency. But also, as the red node strongly inhibits the blue one, an increase in attractants reduce tumbling frequency. The extra slow node (in green) could be said to implement enzyme CheB. With the negative feedbacks to both of the fast nodes, it acts to conpensate the sensitivity of them as the absolute levels of the chemical changes. Finally, the timescales of the fast and slow components in the evolved circuits as well as E. coli’s evolved mechanism are remarkably similar.

Modelling single neurons as several CTRNN nodes

On/off nerve cell detectors in C. elegans

Very recently, it was discovered that individual sensory neurons in C. elegans can detect an increase in salt concentration (while the same neuron on the other side of the body can detect a decrease). They are calling this sensory neurons the ON-cell and OFF-cell for their role in C. elegans’ chemotactic behaviour. I show the behaviour of one of them (the ON-cell, ASEL) during an increase in salt concentration in the figure below. Before this study, a network of neurons was thought to be responsible for such derivation detection, not a single neuron.

Average calcium transients in ASEL neuron in response to NaCl concentration steps of 40 mM from a baseline of 40 mM. The grey band represents standard error of the mean.
Average calcium transients in ASEL neuron in response to NaCl concentration steps of 40 mM from a baseline of 40 mM. The grey band represents standard error of the mean. Taken from Susuki et al. (2008, p. 115). Reproduced without permission.

While a single CTRNN neuron could not display such detection mechanism, if we view the same 3-node CTRNN that we described before as a single neural cell, then a similar behaviour could be obtained (see sketch below).

3-node CTRNN as a single nerve cell
3-node CTRNN as a single nerve cell

Other interesting 3-node component cells

Finally, another successful circuit (and very similar to the one presented above), displayed an additional behaviour that I thought was worth bringing up as well. Basically, as the external input of this 3-node system changes, its output component changes the frequency of its spike-like activity.

Modulation of oscillatory pattern in the output node of a 3-node network for different external inputs to the sensory node.
Modulation of oscillatory pattern in the output node of a 3-node network for different external inputs to the sensory node. In the top part of the figure, example trajectories from the output node are given for three different inputs to the sensory node. In the bottom part of the figure, the pattern of activity of the output node for a range of different external inputs is shown. The shades of gray represent the level of activity of the output node over time.

To summarise this post, there are several different ways to interpret CTRNNs (I won’t go into all of them now). If we choose to interpret them as intracellular signalling components, then using small networks of them as building blocks to build larger networks could be a way to model activity at the level of the neural network (see sketch below).

3-node "neural" network composed where each neuron is comprised of a 3-node CTRNN
3-node "neural" network composed where each neuron is comprised of a 3-node CTRNN

I think this is one promising motivation for using the idea of the building blocks. It is also interesting because it would allow each of the neural cells to have a larger repertoire of dynamics (and thus, to be able to compare them more directly with nerve cells from organisms such as C. elegans).


Alon, U. Surrette, M.C. Barkai, N. and Leibler, S. 1999. Robustness in Bacterial Chemotaxis. Nature. 397.

Alon, U. 2007. An introduction to systems biology: Design principles of biological circuits. Chapman & Hall/CRC.

Beer, R.D. (1995). On the dynamics of small continuous-time recurrent neural networks. Adaptive Behavior 3(4):471-511.

Suzuki H, Thiele TR, Faumont S, Ezcurra M, Lockery SR, Schafer WR (2008). “Functional asymmetry in Caenorhabditis elegans taste neurons and its computational role in chemotaxis.” Nature 454:114-117.


3 thoughts on “Continuous-time recurrent ‘neural’ network as a model of intracellular signalling

  1. Very good summary, my congratulations. But questions:
    I understand that the 3 nodes that form 1 / 3 of a node CTRNN are 3 neurons, correct? What are the parameters of these nodes (threshold, weight, activation function), and the simulation of delays?

  2. Building this modular neural network with the 3-nodes, each having 3 neurons inside is not possible using a CTRNN, just look at your ODE, the ode of the neural network type has no uniform output that you can produce and feed forward as an input to drive another multi-neuronal module. What you get is fit to the experimental data as a readout of the model.
    Have you perhaps managed to solve this issue, by moving your modules into a different NN scheme?

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