Network of Neurons Computational Neuroscience 03 Lecture 6
Slide 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
Slide 3
Use cable theory: R L = r L x/( a 2 ) And multicompartmental
modelling to model propagation of signals between neurons
Slide 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
Slide 5
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
Slide 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
Slide 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
Slide 8
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
Slide 9
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
Slide 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 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
Slide 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)
Slide 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 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
Slide 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
Slide 14
s = 1 s = 4 s = 0.1 Delays the response to the input Also
dependent on starting position
Slide 15
s = 0.1 Filters input based on size of time constant
Slide 16
s = 1 Filters input based on size of time constant
Slide 17
s = 4 Filters input based on size of time constant
Slide 18
Slide 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