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In non linear model of a neuron there was a mention about bias (Bₖ) which was the summation of the synaptic weights.

I want to understand what synaptic bias is and their application/use in neuronal modelling.

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Often artificial neurons are created with conventions that zero is "rest" and 1 is "threshold". The unit starts at 0, and when it reaches 1 it will send an input to all of its targets and be reset back to 0.

This isn't exactly how real biological neurons work, but its a reasonable approximation in some contexts (and can do real world computations).

Most of the inputs to a unit are variable, so you add some voltage to the neuron based on the synaptic weight. For example, one input with a weight of ".2" would move the cell 1/5th of the way to threshold by itself.

When people talk about a bias input, they are referring to an extra "always-on" input with a small synaptic weight. This can maintain a rest greater than 0 in a leaky integrate-and-fire cell; this allows for some more biologically relevant forms of inhibition (negative synaptic weights) to have an influence on the unit, by offsetting the bias input and pushing the voltage back to 0.

More often, I am familiar with using a bias input in networks that are not leaky. These networks just integrate until they reach threshold (although they may or may not get negative inputs). For these networks (or for a leaky network where the bias current is substantial relative to the leak), adding a bias input ensures that the network won't stochastically "fall asleep" due to a period of low activity. If the bias current is equivalent to ".02" units per cycle, for example, you would guarantee that the cell would fire once every 50 cycles with no other input.

There are some real excitable tissues that show similar behavior, but not due to a real bias input. Rather, there are intrinsic channels that slowly depolarize a cell. Some examples are in cardiac myocytes and the sinoatrial node making up the heart's "natural pacemaker", as well as in brain circuitry that controls breathing. The bias input is sort of a "made up" construct that is functionally, but not mechanistically, correlated with real biology.

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