  Pairwise maximum-entropy models have been used in recent neuroscientific
literature to predict the activity of neuronal populations, given only the
time-averaged correlations of the neuron activities. This paper provides
evidence that the pairwise model, applied to experimental recordings, predicts
a bimodal distribution for the population-averaged activity, and for some
population sizes the second mode peaks at high activities, with 90% of the
neuron population active within time-windows of few milliseconds. This
bimodality has several undesirable consequences: 1. The presence of two modes
is unrealistic in view of observed neuronal activity. 2. The prediction of a
high-activity mode is unrealistic on neurobiological grounds. 3. Boltzmann
learning becomes non-ergodic, hence the pairwise model found by this method is
not the maximum entropy distribution; similarly, solving the inverse problem by
common variants of mean-field approximations has the same problem. 4. The
Glauber dynamics associated with the model is either unrealistically bistable,
or does not reflect the distribution of the pairwise model. This bimodality is
first demonstrated for an experimental dataset comprising 159 neuron activities
recorded from the motor cortex of macaque monkey. Using a reduced
maximum-entropy model, evidence is then provided that this bimodality affects
typical neural recordings of population sizes of a couple of hundreds or more
neurons. As a way to eliminate the bimodality and its ensuing problems, a
modified pairwise model is presented, which -- most important -- has an
associated pairwise Glauber dynamics. This model avoids bimodality thanks to a
minimal asymmetric inhibition. It can be interpreted as a
minimum-relative-entropy model with a particular prior, or as a maximum-entropy
model with an additional constraint.
