Dayan & Abbott Week 1

The Single Neuron

Basic Questions/Notes

  1. Figure 1.1 caused me to update my internal image of neurons and their connections. It's now much messier.
  2. Can be over $10^5$ synaptic inputs for 1 axon.
  3. Brain $\approx 10^{11}$ neurons each link to (on average) $\approx 10^4$ others.
  4. Can a neuron have more than one axon? Assuming no, is that a limitation?
  5. "The axon makes an average of 180 synaptic connections with other neurons per mm of length." How far apart can synapses be on a single neuron? Can they be at different points in a longer axon? Are they all clustered at the end?
  6. How does the signal travel down the axon? What is physically happening?
  7. When a synapse is activated and the message passes to the next neuron how does it reset to the starting point and does it exactly?
  8. Firing results are generally noisy even at low levels of processing when presented with the same experience (except less so in audition).
  9. There are both chemical and electrical connections (not sure what chemical connections are and how they affect things but can occur when neurons touch).
  10. Figure 1.2 AP graph. Rest -70mv. Threshold -50mv. Spike 50mv. Spike time 1-2ms. Mainly Sodium (Na+) and Potassium (K+) moving. Intracellular measuring shows voltage change including changes below threshold. Extracellular only shows spikes. Measuring axon only shows spikes (but more detail than extracellular).

Standard Questions/Notes

  1. In the Purkinje cell in 1.1b all the information coming from the dendrites can only lead to a binary firing at any one time. Intuitively this seems like an odd construction. (Though it's more complex if we consider firing rates over time periods).
  2. Is output equal among synapses? Does connecting an extra synapse change output at other synapses?
  3. Is noise in repeated trials due to lack of information (from our side?). Is that what noise means in all cases? Should we consider it as such?
  4. Single neuron can be either excitatory or inhibitory (not both for different synapses). Would it not be useful to sometimes be both?
  5. If talking about firing rates this can mean across a time period or across different trials in the same time bin for the same stimulus.
  6. What is the difference between how neurons 'store' information and how computers store information? Both involve electrical signals.

Technical Questions/Notes

  1. Inputs among the dendrites can be non-linear. Multiple synapses can connect on one dendrite and their combination is not necessarily linear (presumably because they interfere with movement of molecules). To my knowledge this isn't usually modelled. How important is it for brain function.

Spike Trains and Firing Rates

Basic Questions/Notes

  1. Almost always treat action potentials as identical even though they vary in duration/amplitude/shape.
  2. Tuning Curves: The average firing rate as a function of some stimulus. Show how change in some property of the stimulus affects the firing rate.

Standard Questions/Notes

  1. Fig 1.7 example shows neuron responding to retinal disparity is labelled as "an early stage in the representation of viewing distance." This does encode information about the viewing distance of objects, but is that only under the 'correct' interpretation? In general can we say what an encoding represents without knowing how the information is used?
  2. Figure 1.10 is interesting suggesting that double spikes very close together are indicating a stronger stimulus than single spikes and even than double spikes slightly further apart.
  3. It seems like there's no limit in how to combine spiking statistics to find correlations with the stimulus.
  4. Problem (again) is that action potentials are not independent events.

Technical Questions/Notes

  1. Dirac $\delta$ functions are just functions that represent a single spike ignoring the dimensions duration/amplitude/shape such that the integral of the spike is 1.
  2. Neural response function $\rho(t) = \sum_{i=1}^n \delta(t- t_i)$. A series of spikes at each $t_i$.
  3. The firing rate $r(t)$ is the chance for a spike to occur at time t (in reality time $t + \Delta t$ because can't take probabilities of width 0).
  4. $\langle \rho(t) \rangle$ is used for the neural response function when the data is the average over multiple trials with the same stimulus. This gives the probability that the neuron fires over a given short time interval but not the rate of firing over an extended period of time.
  5. The spike count rate $r$ is the amount the neuron fires over a duration. Usually measured in hertz.
    1. $r(t)$ = firing rate = average over many trials for the same stimulus chance to fire at time $t$.
    2. $r$ = spike-count rate = frequency of firing over extended period calculated from one trial.
    3. $\langle r \rangle$ = average firing rate = frequency of firing over extended period averaged over multiple trials.
  6. Problem with the methods depicted in Figure 1.4 is that they measure the spiking both shortly before and shortly after the time. In actual neurons, the firing that will happen shortly after cannot have any effect. Therefore use something like $\alpha$ function: \[\omega(\tau) = [\alpha^2\tau e^{-\alpha\tau}]_+\]
  7. I don't understand the $\alpha$ function equation. As far as I can tell the graph should be similar to $f(x) = xe^{-x}$ but the $f(x) \leq 0$ for all $x$ so the half-wave rectification operation seems to set everything to $0$. As far as I can tell the aim here is to have a shape similar to a Gaussian for times up to $t$ and then nothing after that but this isn't how I understand the equation.