Neurons and Action Potentials

Neurons and Action Potentials

• Zafer İşcan (PhD), an Engineering in Neuroscience, created these notes
• Ramon y Cajal worked on neurons in 1896
• Lichtman & Sanes also worked on neurons in 2007
• Ramon y Cajal was awarded the Nobel Prize in Physiology or Medicine in 1906

Single Neuron Structure

• Dendrites act as input devices, collecting data from other neurons and transmit this data to the soma
• Soma functions as the CPU for nonlinear processing, generating an output signal if the total input exceeds a certain threshold
• Axons act as output devices that deliver the signal to other neurons

Action Potential Dynamics

• Action potentials involve voltage-gated sodium channels and potassium channels, as well as mechanically-gated ion channels
• Researchers have mapped the behavior of all voltage-gated potassium channels

Understanding Spike Trains

• A spike train refers to a chain of action potentials, or spikes, emitted by a single neuron
• The form of the action potential does not carry information since all spikes of a given neuron look alike
• The number and timing of spikes matter
• The action potential is the elementary unit of signal transmission
• The absolute refractory period is defined by the minimal distance between two spikes
• The absolute refractory period is followed by a phase of relative refractoriness where it is difficult, but not impossible, to excite an action potential

Synaptic Transmission

• Synapse refers to the site where the axon of a presynaptic neuron makes contact with the dendrite (or soma) of a postsynaptic cell
• The most common synapse in the vertebrate brain is a chemical synapse
• Synaptic cleft is the tiny gap between pre- and postsynaptic cell membranes
• Action potentials arriving at a synapse starts a complex chain of biochemical processing steps
• These processing steps lead to a release of neurotransmitter from the presynaptic terminal into the synaptic cleft
• Transmitter molecules are detected by specialized receptors on the postsynaptic cell membrane and open specific channels so that ions from the extracellular fluid flow into the cell
• The ion influx leads to a change of the membrane potential at the postsynaptic site, translating the chemical signal into an electrical response
• Postsynaptic potential is the voltage response of the postsynaptic neuron to a presynaptic action potential
• Besides chemical synapses, neurons can also be coupled by electrical synapses

Postsynaptic Potentials (PSPs)

• A postsynaptic neuron i takes input from two presynaptic neurons j = 1, 2
• An excitatory postsynaptic potential (EPSP) is evoked by each presynaptic spike
• An electrode can measure EPSP as a potential difference
• An input spike from a second presynaptic neuron j = 2 arriving shortly after a spike from neuron j = 1 causes a second postsynaptic potential that adds to the first one
• The total change of the potential is approximately the sum of the individual PSPs
• If ui(t) reaches the threshold θ, an action potential is triggered that starts with a large positive pulse-like excursion
• After the pulse, the voltage returns to a value below the resting potential, called hyperpolarization

Spike Response Model (SRM)

• Membrane potential of neuron i is defined by a formula = η(t - tz) + ΣΣ Eij (t-tf) + Urest
• t₁ is the last firing time of neuron i
• €¡j: response of neuron i to spikes of a presynaptic neuron j
• η: form of the spike and the spike afterpotential
• Firing occurs whenever u; reaches the threshold v if ui(t) = v and the rate of change of ui(t) is greater than 0
• The shape of an action potential is typically replaced by a δ pulse in formal models of spiking neurons
• The negative overshoot (spike-afterpotential) after the pulse is included in the kernel n(t – t₁(1)), which accounts for 'reset' and 'refractoriness'
• The pulse is triggered by the threshold crossing at t₁(1)
• Urest is set to 0

Spike Trains

• Spike train of a neuron i is defined by a function S(t) which sums over all firing times
• Spikes are reduced to points in time

Model Limitations

• Limitations include ISI adaptation, bursting neuron activity and inhibitory rebound spikes
• The shape of postsynaptic potentials depends on the level of depolarization and the internal state of the neuron and timing relative to previous action potentials

Neuronal Coding

• The mammalian brain contains > 10¹⁰ densely packed neurons connected in an intricate network
• In every small cortex volume, thousands of spikes are emitted each millisecond
• The UvaNlf neuron (top raster) increases its firing during playback
• During the first few song playbacks, the HVCI neuron produces only a few bursts

Questions about Neuronal Coding

• What is the information in a spatio-temporal pattern of pulses?
• What code do neurons use to transmit that information?
• How do other neurons decode the signal?
• As external observers, can we read the code and understand the message of the neuronal activity pattern?

Coding Problems

• Traditionally, relevant information was thought to be mainly in the neuron's mean firing rate
• The mean firing rate is defined via a temporal average
• Gain function, represents that the output rate v is given as a function of the total input I₀
• Spike Density, PSTH (Peri-Stimulus-Time Histogram) is a measure of average rate over several runs
• Population Activity refers to when a postsynaptic neuron receives spike input from a population
• Population activity is defined as the fraction of neurons that are active in a short interval divided by time

Spike Codes

• The time to first spike is a coding scheme
• A third neuron is the first to fire a spike after stimulus
• The neurons fire with respect to different phases in the background oscillation
• Phase could code relevant information
• Correlations and Synchrony in neuron firings can act as a coding scheme
• Stimulus Reconstruction and Reverse Correlation are ways to identify potential coding schemes

Summary of Neural Signals

• Neuronal signals are composed of short voltage pulses called action potentials or spikes
• These pulses travel along the axon and are distributed to several postsynaptic neurons, therefore evoking postsynaptic potentials
• If a postsynaptic neuron receives several spikes from several presynaptic neurons within a short time window, its membrane potential may reach a critical value, triggering an action potential
• Action potential is the output signal, which is then transmitted to other neurons
• The information that is conveyed from one neuron to the next is contained in the sequence of action potentials
• The problem of neuronal coding is still not fully resolved

Detailed Neuron Models

• Action potentials are the result of currents that pass through ion channels, located near the cell membrane
• Hodgkin and Huxley succeeded in measuring these currents and in describing their dynamics using differential equations
• The Hodgkin-Huxley equations are the starting point for detailed neuron models
• These models account for the many ion channels, different types of synapses, and individual neurons' spacial geometry

Equilibrium Potential

• Neurons are enclosed by a membrane, which separates the cell's interior from the extracellular space
• The concentration of ions inside the cell is different from that in the surrounding liquid
• The difference in concentration creates an electrical potential, which plays a critical role in neuronal dynamics

Nernst Potential

• At thermal equilibrium, positive ions in an electric field will be distributed so that less ions are in a state of high energy and more at low energy. Thus a voltage difference generates a gradient in concentration.
• A difference in ion concentration generates an electrical potential
• This generates the Nernst potential because the concentration inside the neuron is different
• The probability that a molecule takes a state of energy E is proportional to the Boltzmann factor p(E) ∝ exp(−E/kT), where k is the Boltzmann constant and T is the temperature
• p(E) = exp[−q u(x)/kT]
• Ion density at a point
• There are specific proteins embedded in the cell membrane that act as ion gates
• These are the ion pumps and ion channels
• Ion pumps actively transport ions from one side to the other, as a result, ion concentrations in the intracellular liquid differ from that of the surround
• Example values:
  • Na+ concentration inside the cell < than in the surround, ENa ≈ +50 mV
  • K+ concentration inside the cell > than in the surround, EK ≈ -77 mV

Hodgkin-Huxley Model

• The semipermeable cell membrane separates the cell's interior from the extracellular liquid and acts as a capacitor
• If an input current I(t) is injected into the cell, it may add further charge on the capacitor or leak through the channels in the cell membrane
• A battery represents the Nernst potential

Formal Spiking Neuron Models

• Spikes are generated when the membrane potential u crosses some threshold & from below
• The moment of crossing the threshold defines the firing time t(f)
• Integrate and fire model states that A current I(t) charges the RC circuit
• The voltage u(t) across the capacitance (points) is compared to a threshold θ. If u(t) = 0 at time ti(f) an output pulse δ(t − ti(f)) is generated
• Presynaptic spike δ(t − tj(f)) is low-pass filtered at the synapse, therefore generating an input current pulse α(t − tj(f))