This is a very much simplified diagram of a biological neuron. Biological neurons come in a variety of types. There is a lot of further structure and physiology that could be considered. The features of a neuron shown above are those of most interest for those constructing artificial neural networks (other than spiking-neuron-based models, and those relying on synchronous activation, such as the Shastri and Ajjanagadde model: see L.Shastri and V. Ajjanagadde: Behavioral and Brain Sciences (1993) 16, 417-494).
However, from the artificial neural network point of view, a biological neuron operates as follows: electrical pulses from other neurons cause the transfer of substances called neurotransmitters (of which there are several varieties) from the synaptic terminals of a neuron's axon (think "output") across a structure called a synapse to the dendrites of other neurons (call them downstream neurons). The arrival of the neurotransmitter in the dendrite of the downstream neuron increases the tendency of the downstream neuron to send an electrical pulse itself ("fire"). If enough dendrites of a neuron receive neurotransmitters in a short enough period of time, the neuron will fire.
Caveat: neurotransmitter substances may be excitatory or inhibitory. The text above assumes that only excitatory neurotransmitters are involved. Inhibitory neurotransmitters, as the name suggests, reduce the tendency of a neuron to fire. Some neurons have a mixture of excitatory synapses and inhibitory synapses (i.e. synapses using inhibitory neurotransmitters) and will only fire if there is enough additional excitatory input to overcome the effect of the inhibitory synapses.
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