Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics

There is a growing appreciation of the importance of nonlinearities in evoked responses in fMRI, particularly with the advent of event-related fMRI. These nonlinearities are commonly expressed as interactions among stimuli that can lead to the suppression and increased latency of responses to a stimulus that are incurred by a preceding stimulus. We have presented previously a model-free characterization of these effects using generic techniques from nonlinear system identification, namely a Volterra series formulation. At the same time Buxton et al. (1998) described a plausible and compelling dynamical model of hemodynamic signal transduction in fMRI. Subsequent work by Mandeville et al. (1999) provided important theoretical and empirical constraints on the form of the dynamic relationship between blood flow and volume that underpins the evolution of the fMRI signal. In this paper we combine these system identification and model-based approaches and ask whether the Balloon model is sufficient to account for the nonlinear behaviors observed in real time series. We conclude that it can, and furthermore the model parameters that ensue are biologically plausible. This conclusion is based on the observation that the Balloon model can produce Volterra kernels that emulate empirical kernels. To enable this evaluation we had to embed the Balloon model in a hemodynamic input-state-output model that included the dynamics of perfusion changes that are contingent on underlying synaptic activation. This paper presents (i) the full hemodynamic model (ii), how its associated Volterra kernels can be derived, and (iii) addresses the model's validity in relation to empirical nonlinear characterisations of evoked responses in fMRI and other neurophysiological constraints.