In this article, we delve into the dynamics of connectivity fields and establish an effective field formalism to quantitatively compute transitions between different background fields. We employ an eфective action for the connectivity field, which is computed through an expansion of the system’s action around a specific background. This approach enables us to understand the dynamic processes involved in transitioning between different background fields, arising from perturbations in neural activity.
Background Fields
In our study, we investigate the modifications in activities and their influence on the background field of connectivities. The background field defines the typical state of the system when there are no external influences. We focus on understanding how the dynamic processes involved in transitioning between different background fields, which arise from perturbations in neural activity, lead to a new equilibrium defined by the new background field.
Effective Field Formalism
To achieve this, we employ an eфective action for the connectivity field, which is computed through an expansion of the system’s action around a specific background. This expansion captures a situation in which the background field has been perturbed by external influences. Consequently, the actual state of the system at the moment of the transition, which is still characterized by the previous background field, differs from the new equilibrium defined by the new background field and may undergo a transition to a new state.
We leverage our eфective formalism to quantitatively compute these transitions. Within this formalism, the system is collectively described by a field, which is an element of the Hilbert space of complex functions. The arguments of these functions correspond to the parameters used to describe an individual neuron.
Computing Transitions
We use the fact that η2 << 1 to restrict the fields to those of the form:
Z1, ω1 dZ1 dω1
Z (θ, Z) δ
ω−1
(cid:18)
−
(cid:12)
(cid:12)
(cid:12)
(cid:12)
Ψ (θ, Z) δ
ω−1
J, θ, Z, ω−1 dJ dθ dZ
Ω = J (θ, Z) + κ N ω−1
where ω−1 << 1. Using this expression, we can compute the transition probability between two states by integrating over all possible values of ω.
Conclusion
In conclusion, we have presented an effective field formalism for connectivity functions tran-sitions. This approach enables us to understand the dynamic processes involved in transitioning between different background fields, which arise from perturbations in neural activity. By leveraging this formalism, we can quantitatively compute these transitions and gain insights into the underlying mechanisms of brain function.
