1. Network of Neurons Computational Neuroscience 03 Lecture 6

2. 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

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

4. 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 …

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

6.  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

7. 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

8. Much simpler modelling eg don’t 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

9. Can’t 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

10. 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 AP’s, 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

11. 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)

12. Visualise effect of  s as follows. Imagine I starts at some value I 0 and we have sliced time into discrete pieces  t. At n’th time step have: I(n  t) = I n = I n-1 +  t dI/dt Imagining w.r =0 have: Effect of  s Exponential decay

13. 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

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

15.  s = 0.1 Filters input based on size of time constant

16.  s = 1 Filters input based on size of time constant

17.  s = 4 Filters input based on size of time constant

19. 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 <<  s then v = F(I S (t)) pretty quickly so 2 nd model reduces to first. Alternatively if  s <<  r (more usual) we get: Cf leaky integrator, continuous time recurrent nets

20. Models with only one set of dynamics work well for above threshold inputs as low pass thresholding irrelevant, but when signal is below threshold for a while these dynamics become important and both levels are needed

21. For a network replace weight vector by a matrix. Also often replace feedforward input with a vector Feedforward and Recurrent networks Dale’s law states that a neuron can’t both inhibit and excite neurons so wieghts in each row of matrices must have the same sign ie M aa’ (weight from a’ to a) must be +ve or –ve for all a

22. This means that except for special cases M cannot be symmetric since if a’ inhibits a, unless a also inhibits a’ then M aa’ has a different sign to M a’a However, anlaysis of systems is much easier when a is symmetric. Corresponds to making inhibitory dynamics instantaneous. These systems are studied for their analytical properties but systems where excitatory-inhibitory networks have much richer dynamics exhibiting eg oscillatory behaviour

23. Often identify each neuron in a network by a parameter describing an aspect of its selectivity. Eg for neurons in the primary visual cortex can use their preferred spatial phase (ie what angle of line they respond most to) Then look at firing rates as a function of this parameter: v(  r  In large networks there will be a large range of parameters. Assume that the density of each is uniform and equal to p and coverage is dense. Replace the weight matrices by functions W(  ’) and M(  ’) which describe the weights from a presynaptic neuron with preferred angle  ’ to a postsynaptic neuron with preferred angle  we get: Continuous model

24. Pure feedforward nets can do many things and eg can be shown to be able to perform coordinate transformations (habd to body for reaching) To do this they must exhibit gaze dependent gain modulation: peak firing rate not shifted by a change in gaze location but increased

25. Recurrent networks can also do this but have much more complex dynamics than feedforward nets. Also more difficult to analyse Much analysis focuses on looking at the eigenvectors of the matrix M Can show for instance that networks can exhibit selective amplification if there is one dominant eigenvector (cf PCA)

26. Or if an eigenvalue is exactly equal to 1 and others < 1can get integration of inputs and therefore persistent activity as activity does not stop when input stops While synaptic modification rules can be used to establish such precies tuning it is not clear how this is done in biological systems

27. Also can see that recurrent networks exhibit stereotypical patterns of activity largely determined by recurrent interactions and can be independent of feedforwrd input and thus can get sustained activity Input Output Therfore recurrent connections can act as a form of memory

28. Such memory is called working or short term memory (seconds to hours) To establish long term memories idea is that memory is encoded in the synaptic weights. Weights are set when memory is stored. When a similar (or incomplete) feedforward input arrives to the one that created the memory, persistent activity signals memory recall Associative memory: recurrent weights are set so that network has several fixed points which are identical to the patterns of activity representing the stored memories. Each fixed point has a basin of attraction representing the set of inputs which will result in the net ending up at that fixed point. When presented with an input network effectively pattern matches input to stored patterns Can thus examine capacity of networks to remember patterns by analysing stability properties of matrix encoded by synaptic weights

29. Interplay of excitatory and inhibitory connections can be shown to give rise to oscillations in networks Network analysis now problematic so use homogenous excitatory and inhibitory populations of neurons (effectively 2 neuron-groups) and examine a phase plane anlalysis. Can show that non-linearity of activation function allows for stable limit cycles Can also look at stochastic networks where input current is interpreted as a probability of firing: Boltzmann machines. Now need statistical analysis of network properties