Theory of network dynamics

# Theory of network dynamics

Spiking neuron networks and linear response models.

##### Topics covered in this lesson

- Reasons to choose computational neuroscience.
- Linear response dynamics = boiling down complex computations down into to a simpler, mathematically tractable form.
- Linear response functions are known since 1887, formalized by Volterra and Wiener but then mainly for engineering purposes.
- Main idea: perturbing a system in steady state - a small enough perturbation gives a response proportional to the perturbation.
- Transforming stimuli, frequency domain and time domain.
- Green's function = response to a delta perturbation, equivalent to linear response function.
- How measure the linear response function in our system of interest?
- Spike-triggered averages, and how they relate to the Fourier domain.
- Power spectra.
- Receptive fields.
- Dependence on background noise frequency and the firing rate of the neuron - the more neurons spike, the higher frequencies they can encode.
- Why background noise matters to encoding capability.
- Synchrony and pairwise correlations.
- Correlated input. Where can we use linear response?
- Pairwise spike correlation between neurons increases with firing rate.
- Phase transitions between irregular and periodic activity.
- When does the response start to be oscillatory?
- What is its frequency?
- Are perturbations amplified or damped?
- Breaking point for oscillatory activity to appear.
- Linear non-linear models.
- How include higher order kernels without the need to know all of them?

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Prerequisites

Calculus (integration and differentiation), basic linear algebra (matrices, determinants). Some basic transform theory, such as knowing what Fourier transforms do, what a convolution is.

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