Please read this sentence.
Thanks for reading that! Just as you read it, photons emitted by the screen hit your retina, hyperpolarised the photoreceptors, generated an action potential that started to propagate through the optic nerves. If Elon had implanted a biocompatible electrode there, he could have easily snatched your visual input and displayed it on a screen somewhere. Worse yet, if he had it implanted just a little above the optic nerves, right inside the prefrontal cortex, he might have even read into your thoughts! Or maybe not, the technology hasn’t advanced that much yet. But the math sure does exist, and Elon does exist, and so does Neuralink ! So, we better explore the basics before someone pirates our thoughts…
Detecting Action Potentials
Okay, first of all, the first paragraph was unnecessarily dramatised just to grab your attention. If we can ever read into or manipulate thoughts, the first applications will most definitely be in the medical field where it could be of tremendous help to the patients enduring motor impairments.
With that out of the way, we can focus on a single neuron. Now I don’t have a microscope handy, so you gotta trust me, from a macroscopic view Neurons are very simple elements. They receive inputs. If the inputs cross a threshold, they send outputs. That’s all they do.
The inputs come either from bio-transducers called receptors or other neurons. There might be multiple inputs and each input has a weight associated with it. When the total weighted sum, crosses a threshold, a spike potential will be generated and it will propagate through the axon.
This propagation is more of a gallop than a sprint. The axons (elongated parts of the neurons), are ensheathed with insulating fatty myelin that prevents the dissipation of the potential. At regular intervals there exist Nodes of Ranvier which allow the exchange of ions that markedly speed up the transmission of the moving dipole.
As the dipole moves, it creates local changes in ionic concentration and very feeble electrical fields that electrodes can pick up provided someone (and we’re looking at you Elon) manages to place a sensitive enough electrode inside the skull. This can then be analysed with advanced data science.
Normally an action potential consists of an up-curve of depolarisation and a down-curve of repolarisation that overshoots into hyperpolarisation. When observed in bulk, they look like a series of spikes. Now tell me, what mathematical function do you think fits this kinda graph? That is a flat line with occasional spikes.
Modelling Action Potentials
When we discuss mathematical functions, we normally think of exponentials and polynomials. Step functions, for being a little ill-behaved, normally evade our conversations.
A simpler step function would be the Heaviside Step function that steps only at the origin.
If we differentiate this function, we get something peculiar. It’s called a Dirac Delta Function and this can surprisingly come in handy when dealing with Spike Potentials observed during our neuronal transmission.
When there are multiple spikes, we can integrate the delta function over the entire domain of observation and calculate the average using the temporal length of the domain to find the firing rates. We know that action potentials show all or none phenomena and therefore when a receptor is stimulated by a particular stimulus, its firing rate (and not the amplitudes) would increase.
By analysing the firing rates as a function of a given stimulus we can identify which pathways respond to which type of stimulus. Then, all we need is the input from the brain of our subject and without examining the external environment, we’ll find out how the subject perceives the environment.
These delta functions can often be hard to work with. Thus, defining them as a gaussian bell curve with a width tending to zero gives us a workable equation and a framework to play with and work towards understanding the neural code that different parts of the brain communicate with.
So, we are still far away from stealing thoughts. Then again, as recently as in 1989, we could’ve sworn we were far away from communicating with our friend chilling in Antarctica and receiving a reply back with the entire process taking less than even a second. As computational power exponentiates and algorithms become more efficient, we shall eventually start to understand the mysterious ways in which our brain works.