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Network of Neurons Computational Neuroscience 03 Lecture 6

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Connecting neurons in networks Last week showed how to model synapses in HH models and integrate and fire models: Can add them together to form networks of neurons

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Use cable theory: R L = r L x/( a 2 ) And multicompartmental modelling to model propagation of signals between neurons

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However, this soon leads to very complex models and very computationally intensive Massive amounts of numerical integration is needed (can lead to accumulation of truncation errors Need to model neuronsl dynamics on the milisecond scale while netpwrk dynamics can be several orders of magnitude longer Need to make a simplification

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Firing Rate Models Since the rate of spiking indicates synaptic activity, use the firing rate as the information in the network However APs are all-or-nothing and spike timing is stochastic With identical input for the identical neuron spike patterns are similar, but not identical

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Single spiking time is meaningless To extract useful information, we have to average to obtain the firing rate r for a group of neurons in a local circuit where neuron codes the same information over a time window Local circuit = Time window = 1 sec r = Hz

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So we can have a network of these local groups w 1: synaptic strength wnwn r1r1 rnrn Hence we have firing rate of a group of neurons

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Much simpler modelling eg dont need milisecond time scales Can do analytic calculations of some aspects of network dynamics Spike models have many free parameters can be difficult to set (cf Steve Dunn) Since AP model responds deterministically to injected current, spike sequences can only be predicted accurately if all inputs are known. This is unlikely Although cortical neurons have many connections, probability of 2 randomly chosen neurons being connected is low. Either need many neurons to replicate network connectivity or need to average over a more densely connected group. How to average spikes? Typically an average spike => all neurons in unit spike synchronously => large scale synchronisation unseen in (healthy) brain Advantages

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Cant deal with issues of spike timing or spike correlations Restricted to cases where neuronal firing is uncorrelated with little synchronous firing (eg where presynaptic inputs to a large fraction of neurons is correlated) + where precise patterns of spike timing unimportant If so, models produce similar results. However, both styles are clearly needed Disadvantages

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1. work out how total synaptic input depends on firing rates of presynaptic afferents 2. Model how firing rate of postsynaptic neuron depends on this input Generally determine 1 by injecting current into soma of neurons and measuring responses. Therefore, define total synaptic input to be total current in soma due to presynaptic APs, denoted by I s Then work out postsynaptic rate v from I S using: v = F(I S ) F is the activation function. Sometimes use the sigmoid (useful if derivatives are needed in analysis). Often use threshold linear function F=[I S t] + (linear but I S = 0 for I S < t. For t =0 known as half-wave rectification The model

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Although I s determined by injection of constant current, can assume that the same response is true when I s is time dependent ie v = F(I S (t)) Thus dynamics come from synaptic input. This is presynaptic input which is effectively filtered by dynamics of current propagation from synapse to soma. Therefore use: Firing rate models with current dynamics Time constant s If electrotonically compact, roughly same as decay of synaptic conductance, but typically low (milliseconds)

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Visualise effect of s as follows. Imagine I starts at some value I 0 and we have sliced time into discrete pieces t. At nth time step have: I(n t) = I n = I n-1 + t dI/dt Imagining w.r =0 have: Effect of s Exponential decay

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Alternatively, if w.r not 0 Ie it retains some memory of activity at previous time-step (which itself retained some memory of time step before etc etc).Sort of a time average How much is retained or for how long we average depends on s as it governs how quick things change. If its 0 none retained if large lot retained

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s = 1 s = 4 s = 0.1 Delays the response to the input Also dependent on starting position

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s = 0.1 Filters input based on size of time constant

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s = 1 Filters input based on size of time constant

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s = 4 Filters input based on size of time constant

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Alternatively, since postsynaptic rate is caused by changes in membrane potential, can add in effects membrane capacitance/resistance. This also effectively acts as a low pass filter giving: If r