Use in neuroscience[edit]

The Fano factor is used in neuroscience to describe variability in neural spiking. ^[12] In this context, the events are the neural spiking events and the holding times are the Inter-Spike Intervals (ISI). Often, the limit definition of the Fano factor is used, for which,


${\displaystyle F=\lim _{t\to \infty }{\frac {\operatorname {Var} (N_{t})}{\operatorname {E} [N_{t}]}}={\frac {\operatorname {Var} (ISI)}{(\operatorname {E} [ISI])^{2}}}=CV^{2},}$


where ${\displaystyle CV}$ is the coefficient of variation of ISI.


Some neurons are found to have varying ISI distributions, meaning that the counting process is no longer a renewal process. Rather, a Markov renewal process is used. In the case that we have only two Markov states with equal transition probabilities ${\displaystyle p}$, we have that the limit above again converges,^[13] ${\displaystyle F=2{\frac {CV_{2}^{2}\mu _{1}^{2}+CV_{1}^{2}\mu _{2}^{2}}{(\mu _{1}+\mu _{2})^{2}}}+\left({\frac {1}{p}}-1\right){\frac {(\mu _{1}-\mu _{2})^{2}}{(\mu _{1}+\mu _{2})^{2}}},}$ where ${\displaystyle \mu }$ represents the mean for the ISI of the corresponding state.


While most work assumes a constant Fano factor, recent work has considered neurons with non-constant Fano factors.^[14] In this case, it is found that non-constant Fano factors can be achieved by introducing both noise and non-linearity to the rate of the underlying Poisson process.