Iven My Mareels

The Neurodynamics of Epilepsy: Synaptic regulation and reversal potential modulation during seizures in a neural field model with conductance-based synapses

Recent experimental results have shown that during the initiation and termination of epileptic seizures there is a significant change in the micro-ionic environment of cortical neurons [1] that affects network balance between excitation and inhibition. This is evident through non-synaptic changes in the micro-ionic environment such as the excitatory and inhibitory reversal potentials and conductances that have been measured over the course of a seizure in an in vitro mouse model [2]. Although this phenomenon of a change in reversal potentials during seizures has recently been simulated numerically using a relatively small number of detailed multi-compartmental spiking models [3], it has not yet been modelled using larger scale mesoscopic neural field models within an analytical framework. This is because the majority of these models use current-based synapses, which do not take the reversal potentials into account [4] as they are more difficult to incorporate in mesoscopic models of brain dynamics. We present an analysis of a neural field model with conductance-based synapses that takes into account the reversal potentials and the nonlinear multiplicative effect that they have on the associated conductance. The conductance-based synapses model is derived, analysed and juxtaposed with the current-based synapses model and the results interpreted physiologically. A comparative bifurcation analysis of both models reveals that there are significant differences in the oscillatory behaviours that correspond to epileptic seizures [4]. In the conductance-based synapses model, there are endogenous anti-epileptic regulatory or control mechanisms that operate on the synaptic scale, whereas previously these were thought to be mainly on the network level; for example, in terms of feedback, feed-forward and surround inhibition [5]. Further, upon modulation of the reversal potentials, the new model exhibits seizure behaviour that initiates and terminates due to non-synaptic ionic mechanisms, similar to that measured in recent experiments [1]. Seizure dynamics in the brain are modulated by both synaptic regulatory mechanisms and non-synaptic homeostatic mechanisms, which play key roles during seizure initiation, spread and termination [5]. These mechanisms have been investigated using a novel analytical neural field model which has provided insights into understanding epileptic brain dynamics that are not currently observable in electrophysiological experiments and numerical simulations alone.

The neurodynamics of epilepsy: a homotopy analysis between current-based and conductance-based synapses in a neural field model of epilepsy

Unlike Hodgkin-Huxley type spiking models, the overwhelming majority of neural field models use current-based synapses [1]. Although there exist neural field models that employ conductance-based synapses, it is not clear what their exact effects on the dynamics are, particularly with respect to epileptic dynamics. Neural field models of epilepsy typically describe the transition to seizure-like activity as a bifurcation [2]. This research examines the effects of conductance-based synapses on the transition from normal to seizure-like activity in neural field models.

A bifurcation analysis of a modified neural field model: conductance-based synapses act as an anti-epileptic regulatory mechanism

The spread of seizure-like behaviour through the cortex is facilitated not only by hyper-excitable, hyper-synchronous neuronal population firing, but by overcoming the regulatory mechanisms of the brain, such as feedback and feed-forward inhibition. These control mechanisms normally stabilise such pathological behaviour [1]. We suggest an additional network regulatory mechanism in the form of a ‘shunting’ effect based on the properties of conductance-based synapses, an important neurophysiological structure whose mechanism is often overlooked in macroscopic models of brain dynamics. A mathematical neural field model [2] is modified to include conductance-based synapses as opposed to current-based synapses. This is a more realistic description of synaptic dynamics that has a significant effect on the network behaviour [3]. A nonlinear summation of the synaptic currents is introduced that incorporates local feedback from the membrane potential and an ‘effective’ time constant that varies inversely with the amount of input. The result is a more physiologically detailed description of the synaptic current produced by post-synaptic potentials. The fixed-points of the new system are found and a perturbation analysis is performed. The stability of the system is determined and a bifurcation diagram is generated using the external input and network balance as bifurcation parameters. These results are then compared to that of the original model with current-based synapses and the differences interpreted physiologically. The fixed points, dynamics and oscillatory properties of the conductance-based model differ significantly from the current-based model. This is largely due to the ‘shunting’ effect of the synapses, which acts as a network regulatory mechanism. In particular, oscillatory behaviour in the conductance-based model is suppressed. Hence, conductance-based synapses are an important physiological structure whose mechanism of synaptic transmission should not be neglected in mean-field models, particularly when applied to epilepsy.

The effect of network structure on epileptic dynamics: analysis of the synchronisation properties of an inter-network of cortical columns

Focal epilepsy is characterised by the spread of hyper-synchronous seizure activity from pathological cortical tissue (focus) to other parts of the surrounding cortex [1]. Our research will form the basis of a mathematical description of a mesoscopic network of cortical columns, where the network dynamics of seizure-like behaviour will be examined as it spreads from a focal (pathological) column to other columns. Emphasis is on how the local dynamics and the network topology influence the overall global dynamics of the seizure spread. Most of the brain’s connectivity (white matter) is heterogeneous and anisotropic with only the local connections (within a column) being approximately homogeneous. The majority of mesoscopic neural models do not model any spatially heterogeneous or anisotropic structure within the cortex as they quickly become mathematically intractable [2]. The aim of this study is to examine the dynamics of an inter-network of populations of neurons that approximate a heterogeneous inter-network of cortical columns through the structure of a connectivity matrix as opposed to uniform connectivity. Analysis of the behaviour of this inter-network demonstrates the dependence of the dynamics on both the structure of the connectivity matrix and the neural model used either spiking or neural field. The mathematical formalism of complex network theory allows us to examine the relationship between the connectivity and dynamics of a network of cortical columns. By understanding this relationship, the structure of the network can be used to constrain the dynamics so that an order-reduction of a more complicated model can be performed on the network making the model significantly more mathematically tractable. A network of cortical columns is approximated by modelling each column as an area (see Fig. ​Fig.1)1) that has densely connected nodes (intra-population), where each area or column is sparsely connected (inter-population). Singular perturbation methods are used to perform a time-scale separation of the dynamics of the nodes and areas; i.e., the solutions evolve in two different time scales separated by a boundary layer. The time-scale separation can then be used to perform an order reduction of the higher dimensional system into a low-dimensional model that predicts the dynamics of the full model [3]. Epileptic dynamics are examined by analysing the synchronisation of both the intra-population and inter-populations of neurons. The individual nodes synchronise on the fast time scale and these become aggregate nodes on the slow time scale; i.e., the synchronisation within a column compared to the synchronisation between columns. These preliminary results show that the structure of the connectivity matrix has a far greater effect on the dynamics than the type of neural model used, in this case a leaky integrate-and-fire model. Figure 1 [3] This work is the first stage necessary for constructing a physiologically plausible mathematical model of a mesoscopic network of cortical columns that includes more realistic heterogeneous and anisotropic connectivity. Future research will be directed at incorporating an epileptic focus into the network of columns in order to investigate seizure spread. In particular, the relationship between network topology and dynamics will be examined and how this affects the spread of a seizure.

The perturbation response and power spectrum of a mean-field of IF neurons with inhomogeneous inputs

The aim of this study is to construct a bottom-up model of cortical dynamics that is capable of describing the same types of neural phenomena as top-down continuum models, namely the power spectrum, frequency response to perturbation and EEG time-series. The key difference between the two approaches is that the bottom-up approach preserves more of the intrinsic physiological details than the top-down models [1]. A stochastic Fokker-Planck modelling approach is used to describe a network of leak integrate-and-fire (IF) neurons with temporally inhomogeneous inputs. Previous work either calculated the response of a single neuron with conductance-based synapses, or the network with current-based synapses [2]. In this study we use and extend a recently published Fokker-Planck approach [3] within an analytical framework to calculate the dynamical firing-rate of a network with conductance-based synapses receiving temporally inhomogeneous synaptic input. In particular, the network has fully recurrent connectivity with both the steady-state and the dynamic perturbation response of the background activity fed back into the inputs. This is done in a self-consistent formalism [4] for a network of excitatory and inhibitory neurons. The Fokker-Planck formalism enables the calculation of the linear response of the firing-rate to perturbation with recurrent connections. The power spectrum and EEG time-series of the network are calculated by treating the synaptic inputs as an inhomogeneous Poisson process. From this we determine the auto-correlation function, which is identified as a cyclo-stationary process. The signal is then phase-averaged over its period and the Wiener-Khinchin theorem is used to determine the power spectrum from the autocorrelation function. The power spectrum is convolved with a filter to approximate the local field potential propagation through the extra-cellular fluid [5]. The analytical results of the frequency response of the dynamical firing rate and its power spectra are compared with numerical simulation results for a recurrently connected network with conductance-based synapses and temporally inhomogeneous inputs. Results are obtained using parameter values that represent typical cortical in vivo neurons [4]. This work is the first stage necessary for constructing a physiologically plausible mathematical model of a mesoscopic network of cortical columns.

Analysis of the power spectra, autocorrelation function and EEG time-series signal of a network of leaky integrate-and-fire neurons with conductance-based synapses

Address: 1Department of Electrical & Electronic Engineering, The University of Melbourne, Victoria, 3010, Australia, 2The Bionic Ear Institute, 384388 Albert St, East Melbourne, VIC 3002, Australia, 3Department of Clinical Neurosciences, St. Vincents Hospital, Melbourne, VIC, 3065, Australia and 4NICTA VRL, c/Dept of Electrical & Electronic Engineering, University of Melbourne, VIC 3010, Australia