Gamma oscillations are thought to play a role in information processing in the brain. Bursting neurons, which exhibit periodic clusters of spiking activity, are a type of neuron that are thought to contribute largely to gamma oscillations. However, little is known about how the properties of bursting neurons affect the emergence of gamma oscillation, its waveforms, and its synchronized characteristics, especially when subjected to stochastic fluctuations. In this study, we proposed a bursting neuron model that can analyze the bursting ratio and the phase response function. Then we theoretically analyzed the neuronal population dynamics composed of bursting excitatory neurons, mixed with inhibitory neurons. The bifurcation analysis of the equivalent Fokker-Planck equation exhibits three types of gamma oscillations of unimodal firing, bimodal firing in the inhibitory population, and bimodal firing in the excitatory population under different interaction strengths. The analyses of the macroscopic phase response function by the adjoint method of the Fokker-Planck equation revealed that the inhibitory doublet facilitates synchronization of the high-frequency oscillations. When we keep the strength of interactions constant, decreasing the bursting ratio of the individual neurons increases the relative high-gamma component of the populational phase-coupling functions. This also improves the ability of the neuronal population model to synchronize with faster oscillatory input. The analytical frameworks in this study provide insight into nontrivial dynamics of the population of bursting neurons, which further suggest that bursting neurons have an important role in rhythmic activities.

The free-energy principle is a candidate unified theory for learning and memory in the brain that predicts that neurons, synapses, and neuromodulators work in a manner that minimizes free energy. However, electrophysiological data elucidating the neural and synaptic bases for this theory are lacking. Here, we propose a novel theory bridging the information-theoretical principle with the biological phenomenon of spike-timing dependent plasticity (STDP) regulated by neuromodulators, which we term mSTDP. We propose that by integrating an mSTDP equation, we can obtain a form of Friston’s free energy (an information-theoretical function). Then we analytically and numerically show that dopamine (DA) and noradrenaline (NA) influence the accuracy of a principal component analysis (PCA) performed using the mSTDP algorithm. From the perspective of free-energy minimization, these neuromodulatory changes alter the relative weighting or precision of accuracy and prior terms, which induces a switch from pattern completion to separation. These results are consistent with electrophysiological findings and validate the free-energy principle and mSTDP. Moreover, our scheme can potentially be applied in computational psychiatry to build models of the faulty neural networks that underlie the positive symptoms of schizophrenia, which involve abnormal DA levels, as well as models of the NA contribution to memory triage and posttraumatic stress disorder.

Connection strength estimation is widely used in detecting the topology of neuronal networks and assessing their synaptic plasticity. A recently proposed model-based method using the leaky integrate-and-fire model neuron estimates membrane potential from spike trains by calculating the maximum a posteriori (MAP) path. We further enhance the MAP path method using variational Bayes and dynamic causal modeling. Several simulations demonstrate that the proposed method can accurately estimate connection strengths with an error ratio of less than 20%. The results suggest that the proposed method can be an effective tool for detecting network structure and synaptic plasticity.