Week 2: How to Model the Brain = CHECKED Flashcards
McCulluch Pits Model of Neurons 1943 has X1,X2,X3 having synaptic connecitons
to a receiver neuron Y
Simmplest approximation we can make of McCulluch Pits Model of Neurons is
Add inputs of X neurons (X1+X2+X3) which gives output activity of Y neuron
Making MCP model more realistic by saying more inputs more important than others by adding
synaptic weights
Although hardly used, The MCP is the grand father
of all neuron models
Disadvantage of MCP model is that it ignores
properties of ion channels, different types of synapses etc..
We can calculate the output of Y in McCulloh Pits weighted model of neurons by
w1X1 + w2X2 +w3X3 = Y
McCulloh Pits Formula means the large w
influence Y more
Some synapses are more stronger than others due to
learning
We can write w1X1 +w2X2 + w3X3 in McCulloh Pits Model more concisely as realistically there are more than 3 neurons giving input to receiver neuron Y
Writing sigma formula with N = arbitary number of neurons
Transfer function is introducing
one more step between Y and the final output of the neuron
McCulloh Pits Model of Neurons (1943) Transfer function G is… (4)
Define a threshold value Θ
if Y ≥ Θ then Y = 1 (neuron active)
if Y < Θ then Y = 0 (neuron silent)
also called ‘step function’
Final output from McCulloh Pits Model of Neuron is
Y activation of neuron
G(Y) = r = 1 or 0
McCulloh Pits Final Neuronal Model, Y is referred to as
activation of neuron which is fairly abstract notion
McCulloh Pits Final Neuronal Model, Y could be thought of as the internal state of the neuron
in a state that leads to action potentials or does not (neuron is silent)
McCulloh Pits Final Neuronal Model, output is r
it is some measure of output of the neuron given its activation
McCulloh Pits Final Neuronal Model,
We tentatively (not definitely) identify r (the output) with
firing rate (number of action potentials fired per second)
In Linear Neuron Model they do not use a step function as transfer function since
real neurons have a lot of variability in their firing (not just firing just at 0 or 1)
Diagram of Linear Neuron Model Trasnfer Function
Linear Neuron Model’s Transfer function is
piece-wise linear
Linear Neuron Model’s Final Output is (2)
G(Y) = r = Y
r can have values between 0 and infinity (what???)
In Linear Neuron Model transfer function
Y < 0 then neuron is silent because
there can be no negative firing rates so it is off limits
In Linear Neuron Model, it seems unreasonable to have - (2)
firing rate grow without a bound as input increase
We can not have neurons for instance to fire million spikes per second
In Linear Neuron Model, it seems unreasonable to have firing rate grow without a bound as input increase as…
Therefore, in Sigmoid Neuron Model
(2)
Their firing rate can not go faster than a given frequency
We should introduce a saturating transfer function
In Sigmoid Neuron model, (3)
As G(Y) = r grow, Y grows
As G(Y) = r grows more, we hit the threshold where we saturate the output of Y
This transfer function our output does not grow to infinity with infinite inputs
The McCulloh Pits, Linear Neuron and Sigmoid Neuron have different ways in which concept of mapping summed inputs to firing rate due to
having different transfer functions (G)
McCulloch Pitts , Linear and Sigmoid Neuron have same equation of (2)
w1X1 +w2X2+ w3X3 = Y
General form of sigma formala
The McCulloh Pitts, linear neuron and sigmoid neuron models have something in common is that
they have no dynamics
These models have no dynamics
According to these models..
This means… - (3)
According to these models, once Y met threshold value to fire (e.g., Y = 1 in some cases),
This means Y neuron is constantly in a state which it fires action potential
There is no internal mechanisms in these models that changes values of Y to 0 or another value
It costs a lot of metabolic energy to
fire action potentials so it is not possible for neurons to fire action potentials constantly
Since there is no dynamics in these models (McCulloch-Pitts Neuron, Linear Neuron, Sigmoid Neuron) we will have to perform
w1X1 + w2X2 + w3X3 again with different inputs to obtain a new value
These models (McCulloch-Pitts neuron, linear neuron and sigmoid neuron) is still radially different from how
real neurons behave
These models (McCulloch-Pitts neuron, linear neuron and sigmoid neuron) are called
connectionist type models
Connectionist networks are
networks produced with neural models with no dynamics
Dyanmic change in membrane potential integrate and fire model wants to model - (2)
When we raise the membrane potential of a real neuron (below the firing threshold), it will decay back to -70mV over time’
This property is not captured by connectionist models
Integrate and fire model does not mdoel the action potentials as its equation tell us how the MP evolves
with time, given some synaptic inputs and any externally injected currents
Integrate and fire model models the change in membrane potential (dyanmic change) by adding
factors that increase or decrease variable u (membrane potential)
At rest at integrate and fire model, u is
-70 mV (millivolts)
Adding dynamics to integrate and fire model
Factors that increase u is (3)
excitatory synaptic inputs,
injected current
These are positive terms in model’s equation of du/dt
Adding dynamics to integrate and fire model
Factors that decrease u is (3)
We assume this…
at a high u, ion-channels open that bring u back down,
these are negative term — in model’s du/dt equation
We assume this effect is proportional to u. The further away we are from rest (-70mV) the stronger we are pushed back down.
Integrate and fire model equation means:
Change in u over time is equal to –u + the synaptic inputs + any external currents into our neuron
Integrate and fire model adds time constant t, and other variables (urest = resting potential and u is current MP) to make units work out to look like this:
When will tdu/dt (rate of change) be 0? (when will membrane potential have no change - resting membrane potential) - in integrate and fire model
urest - u = 0
To calculate integrate and fire model equation’s we - (3)
spilt the derivative which makes dt not infinitely small but merely very small which makes no more derivative
We can now calculate u2 (u at time t2)from u1 (u at time t1) and all the inputs
Repeat for every neuron in your network given certain connectivity pattern (i.e., specificed by weights) and other inputs
In the integrate and fire model, if the membrane potential hits a threshold value of action potential (e.g., -40 mV) we say that
spike has been fired and then the membrane potential is reset to -70 mV
In integrate and fire model it has dynamics meaning that
once it hits threshold to fire, the membrane potential will eventually decay back to resting membrane potential value (-70mV)
The (leaky) integrate and fire model is also called
‘formal’ spiking neuron model
In the integrate and fire model gives us spike times but the
spike wave-forms are not calculated in this model
The non-spiking relative of IF model (firing rate model)
making changes to IF model (4)
We remove the spiking threshold, the post-spike reset, and u_rest
We re-interpret what u stands for, and (if we want to) we rename it, say to a (activation)
We substitute synaptic action (as an effect on the membrane potential) with the familiar (from connectionist models) summation of incoming inputs
We add a transfer function (from connectionist model), for instance a sigmoid such that negative a values get mapped to 0 and positive values saturate
Firing rate model schematic:
IF and connectionist models
IF and connectionist models
Firing rate model , a is
interpreted as activation of neuron
Firing rate model transfer function turns a into
firing rate
Firing rate model transfer function decays the membrane potential just like the
IF model
Firing rate model captures the
dynamics and does not give spike times
Firing rate model transfer function decays the membrane potential just like the IF model, but we think of it as
firing rate than membrane potential
Am I interested in spike times?
What model?
IF
Am I interested in only care of spikes per second
What model?
firing rate model
Firing rate model has the assumption that the average rate of firing action potentials for a neuron (in response to inputs) adequately
captures the fundamental properties of a neural network
Firing rate model is a non-spiking model meaning (2)
does not model spike
Any phenomena that depends on accurate spike timing can not be modelled with it
Although firing rate model is not spiking,
Firing rate model is a non-spiking model and captures the (2)
dynamic changes in activity (i.e., average rate spikes over time)
Many neural phenomena can be modelled just in terms of rates
The Hodgkin and Huxley model models ion channels and outlines the mechanisms that underline
the propagation and initation of action potentials based on work they did with a squid giant axon
Taxonomy of models
The Hodgkin-Huxley model have an equation
of how each ion channel changes and plug into equation of MP.
General Form Table of Connectionist Neuron Models
