Notes

Content of the Lecture

• What is a Neuronal Spike?
• (Dis)advantages of Digital (spike) vs. Non-Digital Communication.
• Non-Spiking Biological Systems and their Properties.
• Analogue Communication in the Retina, C. Elegans, Locust.
• The Different Types of Action Potentials.
• How Can We Measure Spiking Activity in a Biological Neuron?
• Temporal Coding Schemes with Spikes.
• Deep Learning with "Time to First Spike".
• Introduction to Neuronal Spiking Dynamics.
• Intracellular Integration of Incoming Spikes.
• Action Potential Generation.
• Hodgkin-Huxley Model.

What is a Neuronal Spike

What is a Neuronal Spike? By now most readers should be aware of how an action potential is generated and propagates through a neuron, but here is a short recap.

• Neuron is at resting potential (-70 mV) which is determined by the balance between extracellular and intracellular K+, Na+ and CL- concentrations.
• Neuron receives inputs at its dendrites which raises the membrane potential. If the membrane potential is below the threshold potential (-55 mV), no AP is generated. If the resulting membrane potential is above the threshold, cell depolarization occurs.
• Voltage gated Na+ channels open causing Na+ influx into the cell which raises the membrane voltage which forces neighboring Na+ channels to also open. This continues up until a peak membrane of 40/50 mV.
• The Na+ channels begin to deactivate and voltage gated K+ channels begin to open causing K+ ion efflux. This rapidly lowers the membrane potential to below the baseline/resting potential levels (-80/-90 mV) which is called hyperpolarization. During this stage, the refractory period, the neuron is unable to fire again while the original ion balance/concentration is reestablished via Na+/K+ ATPases.

Digital vs Non-Digital Communication

(Dis)advantages of Digital (spike) vs. Non-Digital Communication There are several disadvantages to sending digital signals along a channel. One of them is quantization loss (as shown in the figure). As we can clearly see, the output waveform (blue) is not perfectly in line with the original analogue input waveform (red). The calculation to determine quanta is straightforward. As an aside, non-linear quantization techniques and schemas exist. One can take advantage of alternative quantization techniques to digitize an analogue signal that has a different entropy, or more information concentrated at lower values at the cost of reducing the resolution in higher values. The most striking argument for analogue activity is given by Shannon's Information Capacity. A calculation done for Crab neurons demonstrates that an analogue channel transfers 2500-6000 bits/s while a digital channel would have 50-220 bits/s: an order of magnitude inferior to an analogue channel. On the other side, the advantage of digital information is that we experience no signal loss during signal transmission across long distances.

Non-Spiking Biological Systems & Different Types of Action Potentials

Analogue Communication in the Retina, C. Elegans, Locust Not every single neuron of every single creature is digital and spiking. There are several examples of species (C. Elegans, Cockroaches, Locusts) and even neuron families in mammals (Photoreceptors, Horizontal Cells, Olfactory Granule Cells) where neurons don't utilize spiking. An entire textbook ("Neurones without Impulses - their significance for vertebrate and invertebrate nervous systems" - Alan Roberts) exists to cover these examples, but the lecture focuses just on one case: C. Elegans. C. Elegans have 302 neurons. None of them spike in the traditional AP generating manner. However, recently, a group found "spiking-like" activity in an AWA (olfactory sensory) neuron type.

After a series of back-and-forth arguments, this evidence proved to be inconclusive and the scientific consensus remains set on the fact that C. Elegans neurons do not generate action potentials, but the resulting "graded" potentials are quire interesting.

An Action Potential is a rapid, all-or-nothing change in the electrical potential across the membrane of a nerve cell or muscle cell. It is triggered by a threshold stimulus, and once it is initiated, it propagates along the cell membrane without decreasing in amplitude.

A Graded Potential, on the other hand, is a change in the electrical potential across the membrane of a cell that varies in amplitude and duration. They are triggered by stimuli that do not reach the threshold level required to initiate an action potential, and their amplitude decreases with distance from the point of stimulation. Graded potentials can be either excitatory or inhibitory and can summate together. In summary, an action potential is an all-or-nothing, rapid change in membrane potential, triggered by a threshold stimulus and propagates without decrease in amplitude. A graded potential is a change in membrane potential that varies in amplitude, triggered by stimuli that do not reach threshold level, and decreases with distance from the point of stimulation.

Why Spikes - from Biology? It has been demonstrated from a signal processing standpoint that digital spikes are inferior to analogue communications. It has also been demonstrated that there are insects that work perfectly well without spikes. So why use spikes? Several reasons:

• Spikes Synchronize Internal Process - A paramecium (single celled organism) uses spikes to forcefully/instantly orient its motile cilia (motors) as part of avoidance behavior.
• Spikes send local information within a Cell - The amoeba uses Mechanosensitive Calcium channels to generate a spike which causes local contraction/compression to generate movement away from an object.
• Calcium spikes regulates homeostatic processes - Ca2+ intracellular signaling cascade/pathways are the foundations of molecular biology. Seeing Ca2+ influx follows an action potential, it would be wasteful to have to come up with a different mechanism to mediate Ca2+ levels. Spiking does this as a byproduct.
• Cells send information across long distances - Any analogue wavelength would decay over time, and be vulnerable to noise fluctuations. Neurons have tricks to reduce this (myelination, increasing axon diameter in sea squids), but ultimately spikes guarantee an intact and reconstructable signal being delivered.
• Energy Efficiency - Self Explanatory. A human uses 100 Watts, the brain takes up to 20 W. Compare that to the power supply unit of a standard desktop (300W0 or a high-end rig (800W).

How To Measure Spiking Activity in a Biological Neuron

How To Measure Spiking Activity in a Biological Neuron? How would one measure a voltage level change of anything? Using a Voltmeter. The only caveat to tis is probe design. Simply placing the tip of the probe in contact with the membrane is usually inefficient (especially at the 1 or 2 micrometer level), thus there are several techniques to properly "clamp" the membrane.

Electrode design is actually quite a hardware challenge for electrical engineers and improvements are being made every year. Alternatively, Neuronal Spiking can be recorded via fluorescence techniques (Voltage Indicators) - Ace1Q-mNeon & Ace2N-mNeon are examples of them.

GEVIs undergo a conformational change in response to a voltage change which changes their fluorescence levels. Finally, there are Calcium indicators that do exactly the same thing but undergo a conformational shift in response toa. Change in intracellular calcium concentration. Advatanges of Imaging:

• Allows to simultaneously record populations of neurons.
• Allows cell type (gene specific) labelling and imaging.
• Enables repeated (chronic) imaging over many months.

Temporal Coding Schemes with Spikes

Temporal Coding Schemes with Spikes Now that we have established that spikes are the method of information transportation in the brain, it is necessary to find a way to decode them. There are several ways in which spiking data can encode information. These can be divided into two primary categories:

• Rate Coding, it refers to information being carried by the firing rate. It is often argued, or assumed, that firing rate captures essentially all relevant information.
• Temporal Coding, it may refer to several quite different ideas:
  • Much of the information may be transmitted by a neuron during certain small intervals of time.
  • Synchronous, or what I would call quasi-synchronous, firing of neurons within and across ensembles may carry important information.
  • The precise timing, or pattern, of spikes may carry information.

In the lecture, several examples were given:

• Spike Rate coding in the Cat V1 encodes direction selectivity. (Cat)
• Phase Coding in Place Cells in the Hippocampus to determine position. (Rat)
• Temporal (Latency) Coding to determine Interaural Time Difference (ITD). (Owl)

Deep Learning with "Time to First Spike"

Deep Learning with "Time to First Spike" An Artificial Spiking Neural Network has been trained to utilize the time to first spike scheme. The (MNIST) 2D image information was encoded as shown in the picture. "Temporal Coding in Spiking Neural Networks with Alpha Synaptic Function", this paper modelled spikes via the Alpha Function. And it can be trained to solve Boolean tasks (AND, OR, XOR), as well as MNIST classification. The alpha synaptic function is a mathematical model that describes the dynamics of synaptic neurotransmitter release. It is characterized by a time constant and a maximum conductance, and it can be used to model the behavior of different types of synapses. In the context of learning with backpropagation, the alpha synaptic function can be used to improve the accuracy of the network by allowing for more precise control of the timing of spikes. By using the alpha synaptic function, the network can more accurately represent temporal patterns in the input data and perform temporal coding.

Neuronal Spiking Dynamics (Hodgkin-Huxley Model)

Hodgkin-Huxley Model Mathematically models neuronal firing. Properties of the Sodium channel:

• Depolarize neuron when open (-70mV +40mV)
• Establish a positive feedback loop (Voltage increase Na+ channels open Voltage increase).

Properties of the Potassium channel:

• K-channels open Voltage decrease.
• K-channels are Voltage-dependent.
• K-channels are slow. (Voltage increase triggers fast Na+ and positive feedback Voltage increase one millisecond later K+ open Voltage decrease).

However, multiple other models (Fitzhugh-Nagumo, Leaky Integrate & Fire, Galves-Löcherback, HTM models) exist and can be derived.