  Mean-field theory links the physiological properties of individual neurons to
the emergent dynamics of neural population activity. These models provide an
essential tool for studying brain function at different scales; however, for
their application to neural populations on large scale, they need to account
for differences between distinct neuron types. The Izhikevich single neuron
model can account for a broad range of different neuron types and spiking
patterns, thus rendering it an optimal candidate for a mean-field theoretic
treatment of brain dynamics in heterogeneous networks. Here, we derive the
mean-field equations for networks of all-to-all coupled Izhikevich neurons with
heterogeneous spiking thresholds. Using methods from bifurcation theory, we
examine the conditions under which the mean-field theory accurately predicts
the dynamics of the Izhikevich neuron network. To this end, we focus on three
important features of the Izhikevich model that are subject here to simplifying
assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions,
and (iii) the distribution of single-cell spike thresholds across neurons.
  Our results indicate that, while the mean-field model is not an exact model
of the Izhikevich network dynamics, it faithfully captures its different
dynamic regimes and phase transitions. We thus present a mean-field model that
can represent different neuron types and spiking dynamics. The model is
comprised of biophysical state variables and parameters, incorporates realistic
spike resetting conditions, and accounts for heterogeneity in neural spiking
thresholds. These features allow for a broad applicability of the model as well
as for a direct comparison to experimental data.
