Random Walks, the Brain Initiative, and the Genius of Einstein's Brain

Over a four-month period in 1905, Einstein published a series of remarkable papers that changed our conception of time and space.


Even more remarkable is the instrument that enabled Einstein to unlock the mysteries of time and space.  His brain.

  

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Some100 billion neurons allowed Einstein to think.  And in order to do this, the neurons in Einstein's developing brain formed a network of neural circuits.  By sending out processes, called axons, each individual neuron made precise connections with other neurons.  

How many circuits do the neurons of a brain make? Which neurons connect together? How is it all arranged?  In fact, nobody knows.  In April 2013, President Obama announced a new research effort, the BRAIN initiative, which has as its goal the discovery of "how the individual cells and complex neural circuits interact in both time and space. "  This huge project strives to develop innovative techniques that can be used to construct a dynamic map of the brain.   

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Of course, this map will beg the question, "how was it built?"  How did each axon know where to find the other neuron that it needed to connect with?  

It is known that there are molecules in the space between neurons that can act as guidance cues for the migrating axons.  By detecting these molecules, the migrating axons are directed towards their target. 

It has long been thought that the guidance cues act as attractants or repellents.  In response to an attractant, the axon directs outgrowth activity towards the side of the axon where the cue was detected.  In response to a repellent, outgrowth activity is inhibited.  A combination of these cues "pulls" and "pushes" the axon to its target. 

In 2008 my laboratory made an interesting observation.  We noticed that a neuron could be tricked into responding to an "attractive" guidance cue.   When we mutated the receptor that the neuron uses to detect the guidance cue, the neuron would send out an axon even though there was no guidance cue.  Moreover, when we looked at many of these neurons we noticed that the direction of axon outgrowth varied.  This was surprising since it meant that if there were no external guidance molecule to provide a directional cue, the neuron would stochastically choose the direction of axon outgrowth.  Until this time, we didn't know that neurons had the ability to choose a direction without a guidance cue. 


This raised a new question, "what if the direction of axon outgrowth is always stochastically chosen?"  That is, there are two responses to a guidance cue.  One response sets in motion a process that turns on axon outgrowth activity that is stochastically directed to any side of the neuron.  A second response is to the external distribution of the guidance cue and it causes the outgrowth activity to be oriented towards the side of the neuron that detects the cue.

Perhaps we never notice that the direction of outgrowth is stochastically determined because the second response always causes the outgrowth activity to be oriented towards the cue.  Practically, this stochastic model is no different than the deterministic “attractive” response model.  In both cases the overall response is to drive the outgrowth towards the side of the neuron that detects the cue.  At this point, it didn't seem that we had a very tractable model.  However our new model proved important because it added a new element into our thinking.  Randomness. 

This new thinking began to surface as we re-examined the effects that certain guidance molecules have on the direction of axon outgrowth from the neuron.  It was known that when certain molecules were removed, the direction of outgrowth could vary.  In the past, the effects of other cues were used to explain this directional variability.  That is, different cues could elicit a response in the axon because the “attractiveness” of one cue had been weakened.  However now I began to wonder if there was a different reason for this behavior.  Perhaps the “ground-state” of the response is actually stochastically directed outgrowth.  Removing the effects of a guidance cue simply makes the decision more random.

But this was paradoxical.  I wanted to know how axons were directed to their targets, but I was hypothesizing that the direction was stochastically determined.  So I began exploring the world of random movement.  What is random movement?  Can it be directed? 

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This brings me back to Einstein and 1905.  One of the papers Einstein published that year was “On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat.”  This landmark paper developed the theoretical groundwork for measurements to confirm the existence of atoms.  In this paper, Einstein gives an explanation for Brownian motion, the irregular motion of particles immersed in a fluid.  It is based on a hypothesis that the particles move about because randomly moving liquid molecules continuously bombard them.  Einstein realized that these impacts, and the movement of a particle over time, could be described probabilistically.

The approach Einstein used is based on a random walk model.  The name “random walk” comes from a challenge to mathematically describe the walk of a drunkard.  Given that the direction of each step of a drunkard is so irregular that the next step can’t be predicted, what is the probability of the drunkard covering a specific distance in a given time?  Einstein essentially adopted the solution to this problem to obtain the probability of a Brownian particle covering a particular distance in time.  Einstein then related the random walk of a single particle to the diffusion of many particles.  

Einstein might have been the first to give a practical explanation of physical phenomena by considering random processes.  This laid a foundation for stochastic modeling.  Besides a Brownian particle, the theory works for the movement of many different types of objects.  Basically, it allows the properties of a macroscopic system to be described even if the behavior of the system is based on the effects of numerous unpredictable, and perhaps unobservable, events.

So what if a migrating axon is actually an inside out Brownian particle?  Whereas the Brownian particle moves because stochastic forces (the impact of randomly directed liquid molecules) are driving it from the outside, the axon moves because stochastic forces (the randomly directed outgrowth activity) are driving it from the inside. 

At any particular time, guidance cues might interact with the axon to determine the probability of axon outgrowth activity happening at a particular side of the axon.  If the cues create a consistent bias over time the axon would be guided in a specific direction. 

Is axon guidance a random walk?  Do random walks underlie how connections are made in the brain?  

Did random walks in the developing brain of Einstein make it possible for us to understand how random walks made it possible for his brain to think about random walks?   

Strikingly, in its quest to map the brain, the new $100 million BRAIN Initiative may be studying the outcome of random walks.  This is amusing since for well over a hundred years random walks have been associated with brain impaired drunkards.

References:

Kulkarni, G., Xu, Z., Mohamed, A., Li, H., Tang, X., Limerick, G., & Wadsworth, W. (2013). Experimental evidence for UNC-6 (netrin) axon guidance by stochastic fluctuations of intracellular UNC-40 (DCC) outgrowth activity Biology Open, 2 (12), 1300-1312 DOI: 10.1242/bio.20136346

Yang, Y., Lee, W., Tang, X., & Wadsworth, W. (2014). Extracellular Matrix Regulates UNC-6 (Netrin) Axon Guidance by Controlling the Direction of Intracellular UNC-40 (DCC) Outgrowth Activity PLoS ONE, 9 (5) DOI: 10.1371/journal.pone.0097258

Xu, Z., Li, H., & Wadsworth, W. (2009). The Roles of Multiple UNC-40 (DCC) Receptor-Mediated Signals in Determining Neuronal Asymmetry Induced by the UNC-6 (Netrin) Ligand Genetics, 183 (3), 941-949 DOI: 10.1534/genetics.109.108654