  The description of neuronal activity has been of great importance in
neuroscience. In this field, mathematical models are useful to describe the
electrophysical behaviour of neurons. One successful model used for this
purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is
composed of two ordinary differential equations. Usually, this model is
considered in his standard formulation, i.e., with integer order derivatives.
In this work, we propose and study the fractal extension of Adex model, which
in simple terms corresponds to replacing the integer derivative by non-integer.
As non-integer operators, we choose the fractal derivatives. We explore the
effects of equal and different orders of fractal derivatives in the firing
patterns and mean frequency of the neuron described by the Adex model. Previous
results suggest that fractal derivatives can provide a more realistic
representation due to the fact that the standard operators are generalized. Our
findings show that the fractal order influences the inter-spike intervals and
changes the mean firing frequency. In addition, the firing patterns depend not
only on the neuronal parameters but also on the order of respective fractal
operators. As our main conclusion, the fractal order below the unit value
increases the influence of the adaptation mechanism in the spike firing
patterns.
