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    Theory of network dynamics

    Difficulty level

    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?

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