Neural Connections

Tyler Sloan is a freelance data artist/scientist. While he is not developing custom Jupyter-based data processing pipelines, he produces computer-generated artwork and data-driven motion design using Open Data and formal scientific models. His artwork combines elements of his training as a developmental neurobiologist (B.Sc, Ph.D.) with his passion for Open Data.

The brain was once thought to be an irreducible mess of fibers; an inter-tangled and arbitrarily-connected reticulum. That was a reasonable assessment using the methods of the day, as a contiguous jumble is all that one would’ve seen from a slice of brain beneath the microscope. With the advent of “la reazione nera” by Golgi, the finite nature and fine morphology of nerve cells could be observed for the first time. Despite Golgi’s continued support, the reticular theory was overturned when, in the hands of Cajal, this tissue staining technique revealed the nervous system was made from what he called discrete nerve elements – and belief in the neuron went mainstream.

Colaterales.
An image of Golgi-stained neurons in the cortex. . Named after the collateral axons branching from the principal axon segment. (photo credit: Lukas Tamayo)

The legacy of the reduced silver nitrate staining was a conceptual reduction of an organ once thought to be irreducible. The philosophical school of thought that dominated following this discovery was simple: by understanding the neuron, one can understand the brain. Images of these preparations show a remarkable calm and tranquility – neurons sparsely floating in seemingly empty space; a minimalist view of the nervous system achieved by systematically under-representing the complexity, by looking only at a small fraction of the neurons.

The impressive progress of neural science in the 20th century carried on in this vein: tissue culture techniques first allowed tissue ‘explants’ to be grown without the confounds the body, and later, for individual cells to be grown without the confounds of the tissue. This further reductionist approach has become a gold-standard in the study of development, used to dissociate between cell-autonomous and non-autonomous processes. Now cell behavior can quantified, generalized, and modelled with math.

Discrete nerve elements
Spinal neurons, dissociated from their tissue and cultured on a glass coverslip for several days.

While there were countless breakthroughs that were only possible by studying cells in a figurative vacuum, by focusing so much on cell-autonomy we miss out on some of the more impressive features of cellular automata. The classic simulations of von-Newmann, Conway, and Wolfram show how elegant complexity can be – how simple rules can lead to emergent, unpredicted patterns. While tissue culture techniques of the modern era can coerce neurons to survive for months in vitro to be studied in isolation, the natural state of the nervous system is primed for self-assembly; neurons working together, often by simple rules, to create something unimaginably complex.

Spontaneous neural networks.
Photomicrograph of a high-density neuronal culture: neurons self-assemble into nodes in a network.

My favorite example of this comes from microscopic observations of neurons left to their own devices. When they are dissociated from their tissue, resuspended and then grown together at high density, they will spontaneously cluster and form networks. This can be seen in my photomicrograph ‘Spontaneous neural networks’. The ‘discrete nerve elements’ of Cajal are primed to self assemble into a semi-functioning nervous system. Nerve cells clump into distinct nodes, forming apparent bridges between one another in collective behavior reminiscent of the swarming of ants and bees, or the twining of vines. A major goal of my SciArt practice is to appreciate and attempt to mimic these patterns and behaviors; to embrace the noise and complexity that make brains so special.

Electric brain in a dish.
Another photomicrograph of high-density neuron culture. This image has a few examples where axonal bridges span a gap with no cells.

In a series of artworks, I wanted to try to understand and mimic some of the behavior I had witnessed when culturing neurons in vitro. While none of these works succeeds to capture all aspects, they each isolate and explore different properties of the self-assembling behavior.

Explants:
A simulation of axon guidance by growth rate modulation – axons are guided towards other explants by growing faster when they’re oriented toward their target.

In ‘Explants’, I started with the stochastic chemotaxis equations from “Axon guidance by growth rate modulation” (Mortimer et al 2010, PNAS). These equations describe mathematically the growth of individual axons in gradients of attractant guidance cues. Instead of simulating a uniform concentration gradient to guide the axons, I assigned each axon at random to be guided towards one of the other explants. Once a connection was made, I simulated axon firing an action potential by drawing Bezier curves between the two partners. Since I ended the simulation before all axons reached their target, the lines are faint, but there was a tendency for a higher density of connections between closely neighboring explants. In more technical terms, the model approaches a fully-connected network, albeit constrained by time. While it doesn’t quite capture the geometry of the networks of real neurons, the residual wandering axons give a glimpse of the imperfect and error-prone nature of axon guidance.

Explants with noisy network topology.
A simulated network of neural connectivity, using a modified nearest-neighbour searching algorithm to define the connections.
(1/3) The first image shows the entire network from a distance.

The next sequence is an exploration of neural circuits at different scales which I called ‘Explants with noisy network topologies’. I started by trying to approximate the kind of geometric patterns I had observed under the microscope between different cell clusters, which more closely resembles a nearest-neighbor topology. I then added some randomness: each axon’s search for the nearest cluster was randomly cut short. That’s to say that they were each provided with imperfect information – each imperfect to varying degrees, thus treading the line between order and disorder.

The next two images in the same sequence were an attempt to give the viewer the experience of progressively zooming in with a microscope; observing the same specimen at different times, with increasing objective magnification. In the second image of the sequence, the connections between clusters are still visible, but we also can see the wandering axons that never found their way and almost fade from view. In the third image, we zoom in further to see a network during its formation with multitude of axons being guided away from their explant, towards their respective targets.

Explants with noisy network topology.
2/3) The second image shows an intermediate zoom on the same network.

Explants with noisy network topology.
(3/3) The third image is a zoom into the axons growing out of a single explant. Axons are guided towards their targets by the growth rate modulation equations.

While I was happy that these simulations succeeded to roughly mimic the appearance of individual wandering axons and the overall network structure, it was not fully satisfying as it used purely top-down processes; using randomly predetermined targets and nearest-neighbor algorithms. So I wanted to create an outgrowth simulation that captured the bundling of axons in a bottom-up manner, to harness the complexity of inter-axonal interactions that must contribute to the self-assembly of the nervous system.

Nerve agents:
An alternative simulation, where axons aren’t guided by any external source. Instead, they are drawn with a multi-agent model exhibiting flocking and trail-following behaviours.

The solution I came up with was to make a multi-agent simulation of outgrowth. This kind of model is used to simulate complex and collective behaviors across many spatial scales, by creating a system of rules for how individual ‘agents’ behave when interacting with one another. I highly recommend trying some online demonstrations using Netlogo to get a sense of how these models work.

I started with some example flocking and trail-following behaviors from the Culebra Behavior Library in Processing.  The first was a 2D experiment I call ‘nerve agents’, wherein I simulated random outgrowth from a neuron cluster to represent a tissue explant. In this experiment, the axons were constrained only by their interactions with one another, not guided by any external source. In this way, I modelled the bundling of axons in terms of the kind of simple interactions that govern the emergent behavior of flocks of birds, or swarms of ants.

What I found particularly interesting about these experiments was that while I had attempted to recreate the bundling of axons growing out from a tissue explant, the results look more like the randomly branching protrusions of filopodia from a cell or growth cone. At a deeper level,  I think this underscores the reason I’m so interested by these kinds of simulations – the apparent scale-invariance of biological growth. This property has been well described in fractal geometry, but I believe this hints at a special characteristic of stochastic biological growth processes that tends to create visually similar structures at very different spatial scales.

The culmination of my explant simulations was to extend the growth and network formation models into 3D with Explants: outgrowth and emergent connectivity’. This was an attempt to combine random outgrowth and collective behavior of axons to explore the emergence of neural connectivity. For these experiments, I used several 3D models of random, bundled axon outgrowth. To approach a similar connectivity as that which I explored earlier in ‘Explants’, I created rules for the axons to stop growing if they reached another explant and to form a connection. The firing of action potentials between these connections are random, but they increase in likelihood as each simulation progresses. As a final touch, to represent the refinement of the connections I added a sub-process by which the connected axons progressively straighten with time, approaching an appearance that more closely resembles the nearest-neighbor topology seen in ‘Spontaneous neural networks’.

Explants: outgrowth and emergent connectivity: Experiments of simulated random axonal outgrowth and emergent, rule-based connectivity. A montage of brief 3D simulations, exploring different growth rules and parameters. Connections occur by chance when axons happen to reach another explant, and the likelihood of firing increases with time.

 

Simulate the neuron, simulate the brain

In keeping with the logical consequences of the neuron doctrine, whereby understanding individual neurons can help lead to understanding the brain, I wanted to take my artistic simulations of individual neuron behavior – and scale them up to represent a functioning brain. A scientific approach would be to take a complex system and to simplify it with statistical methods. For example, by taking raw fMRI data and averaging the signal between subjects and across time. One downstream result of this type of approach is a network wiring diagram that maps out the most important connections, that is human readable and understandable. Fortunately, human readability is not a prerequisite for art, so I decided to take the opposite approach by using a heavily processed connectivity data-set of the human brain, and to reconstruct its complexity.

UrchsBrain:
Simulated connectivity between the 444 resting state functional brain parcels described in (Urch et al 2017, MNI Open Research).

I started with publically available data from the paper: ‘MIST: A multi-resolution parcellation of functional brain networks’ (Urchs et al, 2017). I am very appreciative to the authors for hosting their article and data on the MNI Open Research platform. The piece was namedUrchsBrain’ after the study’s first author, respectfully.

For this piece, I used the connectivity data of functionally-related brain regions at the highest resolution included in the study, containing 444 distinct nodes. Each node was represented as a cluster of neurons, each cluster extended axons to make connections with neurons in other nodes, based on their connectivity data. I simulated firing between cells by drawing curves, paying special attention to the midline crossing to represent the Corpus Callosum. The 2D ‘UrchsBrain’ was intended to be printed on a large canvas with high enough resolution to see individual axons. Additionally, wanted to capture the bustling activity in a video, so I created a dynamic 3D version: ‘This is your brain at rest’.

This is your brain at rest. An animated version of the data from UrchsBrain, with firing between functionally-related regions simulated in 3D. Connectivity determined from fMRI measuring subjects resting-state brain connectivity.

 

Conclusion

It was just over a century ago that Cajal and Golgi shared the Nobel Prize in Physiology and Medicine for the discovery of discrete nerve elements. The past century has seen progress in neurobiology that would have left either man in need of a new pair of shorts. Neurosurgeons can sew together peripheral nerves and 3D-print axon scaffolds. Researchers can witness the ensemble of the axonal prominences of an organism in three-dimensional technicolor, seemingly blurring the boundaries between science and art. Cajal’s annoyance with the inadequacy of the tools of his time to test his chemotactic hypothesis has been countered with a multitude of techniques, generating terabytes of quantitative information of which we are only able to scratch the surface. For this reason, the next hundred years will face a new challenge that is the inverse to that of Cajal’s day; whereas the techniques at his disposal were insufficient to test his theory, soon the theoretical framework will need to be unified to clarify the marvelous, complex assembly of his discrete nerve elements.

Vertebros.
A generative visualization of the location and projection of each of the distinct neuron types in the spinal cord.

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You can see more of my work and keep up date by following me on Twitter, Instagram, or subscribing to my YouTube channel, or through my website: www.Quorumetrix.com

 

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