# Limitations of spike response model and coding

Review of Last Week’s Reading

## Review of Last Week’s Reading

**Keywords**:

- Spiking train: a chain of action potentials (P3)
- Absolute refractory period: minimal distance between two spikes
- synapses: chemical, electrical (gap junctions)
**excitatory and inhibitory**: change of potential due to an arrival spike, positive-> excitatory, negative -> inhibitory**postsynaptic potential**: voltage response- PSP: postsynaptic potential
- IPSP/EPSP: I-> Inhibitory, E->Excitatory

- depolarize/hyperpolarize: increase potential->depolarize, decrease potential->hyperpolarize
- $\text{SRM}_0$: Each response to the incoming spikes are linearly summed up until a spike is triggered. (P7, Equation 1.3)
- $$u_i=\eta(t-\hat t_i)+ \sum_j \sum_f \epsilon_{ij}(t-t_j^{(f)}) + u_{\mathrm{rest}} $$

- adaptation: fig1.5
- regular firing neurons/fast-spiking neurons/bursting neurons: fig 1.3. fast-spiking is what would SRM0 give us
- rebound spikes: fig 1.3 D

## Limitations (cont’d)

### Saturating Excitation and Shunting Inhibition

**Facts**: shape of PSP depends on

- level of potential,
- internal state of the neuron, like state of ion channels.

- The PSP depends on the potential of the neuron itself when the presynaptic spike arrives.
Interesting facts:
- IPSP, usually leads to hyperpolarization. Amplitude is larger if the neuron has a higher potential when presynaptic spike arrives. However, the response is reversed, i.e., depolarizing, if the potential is already hyperpolarized a lot. Reason for a reversal response is that PSC switch to the other direction if the potential $u_0$ before spike is due to the wrong direction of the PSC.
- EPSP is larger given lower potential, i.e., more depolarization.

**PSC**: postsynaptic current, which is proportional to the actual effective potential (**reversal potential**) on the neuron membrane.- Shunting inhibition: A few inhibitory synapses shunt input of a few hundred excitatory synapses.

## Neuronal Coding

One of the ways to present neuron spikes is the **spatio-temporal patterns** of pulses. (Fig 1.8)

**mean firing rate**: $\nu= \frac{n_{\mathrm{sp}}(T)}{T}$, i.e., number of spikes during time $T$ divided by the time span $T$.- History:
- Adrian, 1926, 1928: firing rate of stretch receptor neurons -> forces on muscle

- Temporal average: is an approximation by filtering a lot of information out.
- Objections:
- brain activities are fast, too fast to allow coding in temporal average. 400ms for example.
- evidence of temporal correlations betwen pulses of different neurons.
- stimulus-dependent synchronization of activity in many neurons

### Rate Codes

**Mean firing rate**:

temporal averaging (fig 1.9) (Boltzmann?):

- $\nu=\frac{\text{No. of spikes during time }T}{\text{time }T}$.
- Usually $T\sim 100\mathrm{ms}$ or $T\sim 500\mathrm{ms}$.
- The general principle is to increase the time until no change in the average during a single stimulus.
- Example: leech touch receptor,stronger the stimulus -> more spikes during a $500\mathrm{ms}$ average.
- Working if

- mapping of one input variable to a single output result, i.e., $\mathrm{single output}=f(\mathrm{single input})$. Example: force on receptors -> average firing rate.
- stimulus is not fast-varying
- doesn’t require fast reaction

- Evidence for other types of coding:

- Fly react in 30-40ms
- Human react to some visual in a $\sim 10^2$ms
- Human can detect images even the images was shown for 14-100ms.

averaging over repetitions of experiments (fig 1.10) (partial ensemble average, Gibbs?):

- many runs of experiment + PSTH (fig 1.10)
- PSTH: peri-stimulus-time histogram, peri means we count the spikes in a time interval $\Delta t$.
**Spike density**: $\rho(t) = \frac{n_K(\text{from } t \text{ to }t+\Delta t)}{K}/\Delta t$, where $K$ is the number of runs done, while $n_K(\text{from } t \text{ to }t+\Delta t)$ is the sum of all spike of all runs from time $t$ to $t+\Delta t$. If we have enough runs we could take $\Delta t\to 0$ to get the continuous density.- Unit of spike density: Hz.
- Meaning of this average: not really what happens but only ensemble average. It reflects some properties of the pattern but not what is done in the brain.
- Ensemble average is more or less equivalent to time average if the stimulus is constant.
- Ensemble average is almost identical to the population average if a lot of identical neurons are firing independently.

averaging over populations of neurons (fig 1.11) (another ensemble average):

- Population activity: $A(t) = \frac{\text{No. of spikes during time } t\sim t+\Delta t \text{ of all the }N \text{ neurons that has the same input}}{N}/\Delta t$
- No. of spikes during time $t\sim t+\Delta t$ of all the $N$ neurons that has the same input $n_{act}=\int_t^{t+\Delta}dt \text{ all spike written in delta function}=\int_t^{t+\Delta}dt \sum_j\sum_f \delta(t-t_j^{(f)})$.
- However,

- no actual homogeneous identical neurons

- Inhomogeneous? Population vector coding. $i$th neuron takes input $x_i$.
- Given a vector input, what is the average position (location on the vector index) of all neurons that gives output? $\mathrm{Average Location of Stimulated Neurons}=\frac{\int_t^{t+\Delta t} dt \sum_i \sum_f x_j \delta(t-t_j^{(f)}) }{\int_t^{t+\Delta t} dt \sum_i \sum_f \delta(t-t_j^{(f)}) }$.
- Used on neuronal activiy in primate motor cortex. <- Not sure how this is done.

### Spike Codes

- Time-to-first-spike, a Simple Scheme of Spiking Codes: fig 1.12. Three neurons which are responsible for reading a picture of three pixels of three different color for example, will spike sequentially if each of them are responsible for a different color.
- Assuming each neuron is inactive after its spike <- due to some mysterious inhibition: this is called
**time-to-first-spike**

- Assuming each neuron is inactive after its spike <- due to some mysterious inhibition: this is called
- Coding by phase (fig 1.13): periodic spikes (common in hippocampus etc), relative phase between neurons (or relative to oscillatioins of background periodic spikes) could carry information.
- what is background oscillation? fig 1.14

- Example of sychronization and information
- Reverse correlation approach: investigate the averaged input instead of averaged PSTH (as we have done in Fig 1.10). So we find what input is required to generate a spike. Given spike train we can estimate the input by summing up the input of each spike! Fig 1.16B. Eqn 1.12.

### Relation Between Spike Codes and Rate Codes

- Define a rate $\nu$ given a spike code, $\nu(t)=\frac{\text{effective number of spikes during time T}}{T}=\int d\tau K(\tau)S(t-\tau)/\int d\tau K(\tau)$, where $K(\tau)$ is window function which gives us the time interval of measurement, and $S(\tau)=\sum^n_{f=1}\delta(t-t^{(f)})$. So $\int d\tau K(\tau)S(t-\tau)$ is the number of spikes during the time window given by $K$.
- How to reconstruct rate code? If eqn 1.12 is true (linear reverse correlation approach is true) -> rate code, otherwise -> spike code.
- As an example, use Volterra expansion and drop the information about the reconstructed input correlations between two spikes.

OctoMiao, OctoMiao Another Person (2016). 'Limitations of spike response model and coding', Connectome, 03 April. Available at: /snm/limitations-srm-contd-and-coding/.

## Table of Contents

**References:**

**Current Ref:**

- snm/limitations-srm-contd-and-coding.md

**Links from:**

###### Spiking Neuron Models Reading Club

Introduction to reading club of spiking neuron models, schedule, and notice

###### Single Neuron Models

Neuron biological properties

###### Guass-Markov Theorem and Multiple Regression

Gauss-Markov Theorem

###### Limitations of spike response model and coding

Review of Last Week’s Reading

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