  We investigate a mutual relationship between information and energy during
early phase of LTP induction and maintenance in a large-scale system of
mutually coupled dendritic spines, with discrete internal states and
probabilistic dynamics, within the framework of nonequilibrium stochastic
thermodynamics. In order to analyze this computationally intractable stochastic
multidimensional system, we introduce a pair approximation, which allows us to
reduce the spine dynamics into a lower dimensional manageable system of closed
equations. It is found that the rates of information gain and energy attain
their maximal values during an initial period of LTP (i.e. during stimulation),
and after that they recover to their baseline low values, as opposed to a
memory trace that lasts much longer. This suggests that learning phase is much
more energy demanding than the memory phase. We show that positive correlations
between neighboring spines increase both a duration of memory trace and energy
cost during LTP, but the memory time per invested energy increases dramatically
for very strong positive synaptic cooperativity, suggesting a beneficial role
of synaptic clustering on memory duration. In contrast, information gain after
LTP is the largest for negative correlations, and energy efficiency of that
information generally declines with increasing synaptic cooperativity. We also
find that dendritic spines can use sparse representations for encoding of
long-term information, as both energetic and structural efficiencies of
retained information and its lifetime exhibit maxima for low fractions of
stimulated synapses during LTP. In general, our stochastic thermodynamics
approach provides a unifying framework for studying, from first principles,
information encoding and its energy cost during learning and memory in
stochastic systems of interacting synapses.
