Part 2. The proposed Neuron model

2.1.    Description of the neuron model used

• Each neuron N has a threshold θ_n
• Each neuron N receives input signal of intensity i_n
• When the input signal intensity i_n is greater than or equal to the neuron's threshold θ_n , then the neuron fires with output signal intensity O_n

2.2.    In this neuron model, there are two types of neurons :

     (i) Binary neurons , which produce only binary output when activated;


If i_a ≥ = θ_a , then O_a = 1 ; else O_a = 0

     (ii) Non-binary neurons , which produce output signal intensity equal to input when activated;


If i_a ≥ = θ_a , then O_a = i_a ; else O_a = 0


     Binary neurons are represented as circles while non-binary neurons are represented as squares.

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2.3.   Connection weights

• Connections may exist between two neurons and the strength of the connection is termed as 'connection weight'
• If neuron A has a downstream connection to neuron B , the 'connection weight' from A to B can be represented as w_ab
• When the neuron A sends an output signal O_a to neuron B, the signal is amplified by a factor w_ab
  
• When the input signal intensity i_b is greater than or equal to the neuron B's threshold θ_b , then neuron B fires
  
•      ie., the input signal to neuron B is :     i_b = O_a * w_ab
  

Connections in binary neurons :


Input signal to Neuron B = output signal from A * ( connection strength from A to B)
ie., i_b = O_a * w_ab
If i_b ≥ θ_b , then O_b = 1 ; else O_b = 0

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Connections in non-binary neurons :


Input signal to Neuron B = output signal from A * ( connection strength from A to B)
ie., i_b = O_a * w_ab
If i_b ≥ θ_b , then O_b = i_b ; else O_b = 0

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2.4.   Excitatory and Inhibitory connections

•   The signal from one neuron to another could be either excitatory or inhibitory.
•   An incoming inhibitory signal makes a neuron less likely to fire.
•   Excitatory connections have positive connection weights ,for example :   w_ab = 0.8
•   Inhibitory connections have negative connection weights ,for example :  w_ab = -1



Inhibitory connections :

Let Neuron B’s firing threshold θ_b = 1
An excitatory connection w_ab exists from neuron A to neuron B with connection weight 1
An inhibitory connection w_cb exists from neuron C to neuron B with connection weight -1
If neuron A and neuron C fire at the same time , with output intensity 1 each, then :
Input to neuron B : i_b = ( O_a * w_ab ) + ( O_c * w_cb)
                        i_b = ( 1 * 1 ) + (1 * -1 )
i_b = 0
(The excitatory signal from A is cancelled out by the inhibitory signal from C)
Since i_b < θ_b , neuron B doesn’t fire.

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2.5.   Synaptic steps in a Firing sequence

•   In a neural circuit, all neurons do not fire at the same time.
•   They fire in a sequential order, depending on the way they are connected to each other.
•   The upstream neurons fire first, followed by the downstream neurons.


Firing sequence : example 1

Sequence of firing :
Neuron A fires , then after a few milliseconds B fires , then after a few milliseconds C fires .
The signal from A reaches B in one "synaptic step"
The signal from A reaches C in two "synaptic steps"




Firing sequence : example 2 :

Sequence of firing :
Neuron A fires , then after a few milliseconds B & C fire at same time , then D fires, then E fires .
So it has taken 'three steps in time'(synaptic timesteps) for the signal from A to reach E
(assuming input to A,B,C,D,E cross their corresponding thresholds)

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2.6.   Temporal connections : Timed delay in neurotransmission

•   In this model, we introduce connections that incorporate a precisely timed delay in neural-signal transmission.
•  A signal through a t+1 connection reaches the target neuron in one synaptic timestep
•  A signal through a t+2 connection reaches the target neuron in two synaptic timesteps.
•  A connection can have upto t+H connections, where H is the maximum delay allowed for signal transmission for that neural circuit.
Timed delay in neurotransmission : example 1


Timed delay in neurotransmission : example 2

Sequence of firing :
Neuron A fires , then after a few milliseconds B fires , then after a few milliseconds C & D fire at same time .
Since there is a t+2 connection from A to C, the signal from A reaches C at the sametime the signal from B reaches C

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2.7.   Firing frame : Representing neuron states over time

•   To help with visualizing the firing states of neurons over time, we can tabulate the output signals of neurons over time.
•   In this table, which we term as "Firing Frame", we have time in Y axis and neuron names in X axis
•   Each cell represents the output signal intensity of a neuron at specific time


Firing frame : example 1 :

In this neural circuit composed of binary neurons, (assuming that firing thresholds are met)
the sequence of firing is : A fires first, then B & C fire at same time, then D fires , and then E fires.
The output of each of these binary neurons is 1 , which is entered in the cell corresponding to the time the neuron fired.





Firing frame : example 2 :

In this neural circuit, note that the connection from A to C is a t+2 connection,
so it takes two synaptic steps for the signal from A to reach C ,upon which C fires.

Index

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