Biol Cybern DOI 10.1007/s00422-014-0616-4
ORIGINAL PAPER
Bifurcation and oscillation in a time-delay neural mass model Shujuan Geng · Weidong Zhou · Xiuhe Zhao · Qi Yuan · Zhen Ma · Jiwen Wang
Received: 20 May 2013 / Accepted: 17 June 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract The neural mass model developed by Lopes da Silva et al. simulates complex dynamics between cortical areas and is able to describe a limit cycle behavior for alpha rhythms in electroencephalography (EEG). In this work, we propose a modified neural mass model that incorporates a time delay. This time-delay model can be used to simulate several different types of EEG activity including alpha wave, interictal EEG, and ictal EEG. We present a detailed description of the model’s behavior with bifurcation diagrams. Through simulation and an analysis of the influence of the time delay on the model’s oscillatory behavior, we demonstrate that a time delay in neuronal signal transmission could cause seizure-like activity in the brain. Further study of the bifurcations in this new neural mass model could provide a theoretical reference for the understanding of the neurodynamics in epileptic seizures. Keywords Neural mass model · Electroencephalography · Bifurcation · Spike-like oscillation · Seizure
S. Geng · W. Zhou (B) · Q. Yuan · Z. Ma School of Information Science and Engineering, Shandong University, 27 Shanda Road, Jinan 250100, People’s Republic of China e-mail: [email protected] S. Geng School of Information and Electric Engineering, Shandong Jianzhu University, Jinan 250100, People’s Republic of China X. Zhao · S. Geng · Q. Yuan · W. Zhou · Z. Ma · J. Wang Suzhou Institute, Shandong University, Suzhou 215123, People’s Republic of China X. Zhao · J. Wang Qilu Hospital, Shandong University, Jinan 250100, People’s Republic of China
1 Introduction An electroencephalogram (EEG) is produced by amplifying and recording voltage differences between electrodes, placed on the scalp or cerebral cortex, which reflect the electrical activity in the brain. EEGs can often be decomposed into distinct frequency bands (delta, 1–4 Hz; theta, 4–8 Hz; alpha, 8–12 Hz; beta, 12–30 Hz; gamma, 30–70 Hz). EEGs are an important tool to study human brain activity in general, and epileptic processes in particular, with the appropriate time resolution. In neurology, a chief application of EEGs is used as a diagnostic for epilepsy, since epileptic brain activity can produce unambiguous abnormalities in a standard EEG study. In order to simulate the electrical activity of the brain with its intricate cortical structures, various mathematical models have been developed (Deco et al. 2003). One class of such models is known as neural mass models (NMMs), which simulate the dynamics of neuronal populations by providing a balance between mathematical simplicity and biological plausibility. They are currently widely used as generative models to explain noninvasive electrophysiological brain measurements such as those found in EEGs. NMMs describe neural function at a mesoscopic level (Deco et al. 2003; Coombes 2010), in contrast to single neuron ‘integrate-and-fire’ models (Abbott 1999), or the more elaborate Hodgkin and Huxley models (Hindmarsh and Rose 1984; Hodgkin and Huxley 1952). NMMs quantify the mean firing rates and mean postsynaptic potentials of distinct neuronal populations, called neural masses (NMs). While single neurons are considered to be the primary computational units of the brain’s architecture (Hodgkin and Huxley 1952; Finger 2001), it is also widely accepted that the relevant information processing underlying brain function in both healthy and diseased states is carried out by ensembles of interacting neu-
123
Biol Cybern
rons at the mesoscopic level (Shepherd 1991; Shepherd et al. 1975; Jirsa 2004). The first NMMs were proposed in the early 1950s (Beurle 1956). Substantial contributions were made by Freeman and co-workers as part of their study of perceptual processing in the olfactory system (Freeman 1978; Eeckman and Freeman 1991). Freeman underscored that the simplicity and generality of the elements comprising his model make it applicable to other neural systems (Freeman 1987). Similar ideas were developed by Lopes Da Silva et al. and led to the design of a lumped-parameter model, which was able to generate an alpha rhythm (Lopes da Silva et al. 1974, 1976). Starting with the lumped-parameter model, Jansen et al. developed a biologically inspired mathematical framework for simulating spontaneous electrical activities of neuron assemblies as recorded by an EEG, with a particular emphasis on alpha activity (Jansen et al. 1993; Jansen and Rit 1995). The framework has been used to explain epilepsy-like brain activity (Wendling et al. 2000), as well as various narrow-band EEG oscillations, ranging from the delta to the gamma frequency bands (David and Friston 2003). Nevado-Holgado et al. (2012) developed a multi-objective genetic algorithm that can estimate parameters of a neural mass model from clinical EEG recordings. Estimating the parameters of neural mass models generally involves the Bayesian inversion of dynamic causal models using standard variational system identification techniques. Much scientific work has been devoted to the understanding of the behavior of neural mass models. Examples include the derivations of stability conditions of the synchronized state in arbitrary networks (Belykh et al. 2005; Hennig and Schimansky-Geier 2008) as well as the derivations of the equations for the synchronization manifold (DeMonte et al. 2003; Jirsa 2008) for heterogeneous networks. More general, when the system exhibits strong deviations from the synchronized state, mode decomposition techniques become more appropriate (Assisi et al. 2005; Stefanescu and Jirsa 2008). The bifurcation theory of nonlinear systems is a useful tool for investigating the dynamical behavior in neural mass models. For example, Grimbert and Faugeras (2006) used bifurcation analysis to investigate the effect of the input to a NMM on the dynamics of the model and found that rhythmic activities such as alpha rhythm and epileptic wave are related to the structure of a set of periodic orbits and their bifurcation. Touboul et al. (2011) proposed a general framework for studying the bifurcations of neural mass models defined by ordinary differential equations. Breakspear et al. (2005) explained the essential difference between the topology of the global bifurcation diagram of absence and tonic-clonic seizures. NMMs are now used routinely in the dynamic causal modeling of invasive and noninvasive electrophysiological time
123
series (David et al. 2006). In this context, NMMs are used as forward models of hidden neuronal states (that are then mapped to sensors). This provides a complete generative model of observed electrophysiological signals. Dynamic causal modeling (DCM) has been applied to numerous paradigms in the cognitive neurosciences using both EEG, MEG (David et al. 2006; Garrido et al. 2007; Kiebel et al. 2008, 2009), and LFP recordings (Moran et al. 2008). Generally, dynamic causal models use multiple NMMs that are coupled with extrinsic connections. Furthermore, they incorporate explicit delay parameters accounting for within (intrinsic) and between (extrinsic) neural mass connections (Kiebel et al. 2006). In this paper, we focus on a single neural mass with multiple populations and provide a quantitative bifurcation analysis of intrinsic temporal delays. Animal studies suggested the existence of multiple thalamocortical and corticothalamic pathways, yielding a range of thalamocortical delays from 0.1 to 5 ms and longer corticothalamic delays from 1 to 30 ms in the mouse (Salami et al. 2003; Liu et al. 2001), rat (Sawyer et al. 1994; Beierlein and Connors 2002), rabbit (Swadlow and Weyand 1981; Swadlow 2003), cat (Wilson et al. 1976; Tsumoto et al. 1978; Tsumoto and Suda 1980; Miller et al. 2001; Sirota et al. 2005), and macaque monkey (Briggs and Usrey 2007). Scaling the above data to human brain size (Robinson et al. 2004), one can estimate delays of roughly 10 ms for the human thalamocortical pathway and a few tens of milliseconds for the corticothalamic pathway. The work of Roberts and Robinson (2008) indicated that seizure waveforms in the cortex and thalamus depend on the axonal and synaptodendritic delays. Previous work of models of brain has not explicitly taken into consideration delays in connections. Although delays make the analysis more difficult, their effects may be important for the applications. Research on Hopfield-type neural networks with delays first introduced by Marcus and Westervelt (1989) has also shown that delay can modify the global dynamic in interesting ways. Transmission delay seems to play a significant role in integration of information arriving to a single neuron in different spatial and temporal windows and also at the network level in interneuron communication. The theoretical study of the dynamics of simple units organized into networks with delayed couplings revealed a rich variety of possible scenarios of transition to a global oscillatory behavior induced by the delay (Bungay and Campbell 2007; Campbell et al. 2005; Guo 2005, 2007; Song et al. 2005; Yuan and Campbell 2004; Yuan 2007; Wei and Velarde 2004). The emerging oscillations can exhibit different patterns sensitive to the delay. Since then, delays have been inserted into various simple neural networks. Campbell et al. (2004) investigated the stability and bifurcations in the delayed neural network of two coupled three-neuron subnetworks. In this work, we modify the model proposed by Jansen et al. by adding a time delay in the inhibitory feedback loop
Biol Cybern
to simulate signal transmission among neurons, resulting in a neural mass model with time delay. The new model is able to reproduce several different types of EEG activities including alpha wave, interictal EEG, and ictal EEG, with all parameters (excepting extrinsic inputs) being kept constant and equal to standard values. Through an analysis of the influence of the time delay on model’s dynamical behaviors, we succeed in showing that a transmission delay can make the model to produce seizure-like activity.
2 Model description As shown in Fig. 1, the model consists of three subsets of neurons, namely the main cells (i.e., pyramidal cells), the excitatory interneurons, and the inhibitory interneurons. The influence of neighboring areas is represented by an excitatory input p(t) (modeled by band-limited white noise) that globally describes the average density of afferent action potentials. The model output y(t) corresponds to the sum of postsynaptic potentials in activated pyramidal cells, which can in turn be used to synthesize EEG activity. In our model, each subset of neurons involves two operators. The first transforms the average pulse density of action potentials into an average postsynaptic membrane potential, described by the response functions given by Aate−at t ≥ 0 (1) h e (t) = 0 t
ORIGINAL PAPER
Bifurcation and oscillation in a time-delay neural mass model Shujuan Geng · Weidong Zhou · Xiuhe Zhao · Qi Yuan · Zhen Ma · Jiwen Wang
Received: 20 May 2013 / Accepted: 17 June 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract The neural mass model developed by Lopes da Silva et al. simulates complex dynamics between cortical areas and is able to describe a limit cycle behavior for alpha rhythms in electroencephalography (EEG). In this work, we propose a modified neural mass model that incorporates a time delay. This time-delay model can be used to simulate several different types of EEG activity including alpha wave, interictal EEG, and ictal EEG. We present a detailed description of the model’s behavior with bifurcation diagrams. Through simulation and an analysis of the influence of the time delay on the model’s oscillatory behavior, we demonstrate that a time delay in neuronal signal transmission could cause seizure-like activity in the brain. Further study of the bifurcations in this new neural mass model could provide a theoretical reference for the understanding of the neurodynamics in epileptic seizures. Keywords Neural mass model · Electroencephalography · Bifurcation · Spike-like oscillation · Seizure
S. Geng · W. Zhou (B) · Q. Yuan · Z. Ma School of Information Science and Engineering, Shandong University, 27 Shanda Road, Jinan 250100, People’s Republic of China e-mail: [email protected] S. Geng School of Information and Electric Engineering, Shandong Jianzhu University, Jinan 250100, People’s Republic of China X. Zhao · S. Geng · Q. Yuan · W. Zhou · Z. Ma · J. Wang Suzhou Institute, Shandong University, Suzhou 215123, People’s Republic of China X. Zhao · J. Wang Qilu Hospital, Shandong University, Jinan 250100, People’s Republic of China
1 Introduction An electroencephalogram (EEG) is produced by amplifying and recording voltage differences between electrodes, placed on the scalp or cerebral cortex, which reflect the electrical activity in the brain. EEGs can often be decomposed into distinct frequency bands (delta, 1–4 Hz; theta, 4–8 Hz; alpha, 8–12 Hz; beta, 12–30 Hz; gamma, 30–70 Hz). EEGs are an important tool to study human brain activity in general, and epileptic processes in particular, with the appropriate time resolution. In neurology, a chief application of EEGs is used as a diagnostic for epilepsy, since epileptic brain activity can produce unambiguous abnormalities in a standard EEG study. In order to simulate the electrical activity of the brain with its intricate cortical structures, various mathematical models have been developed (Deco et al. 2003). One class of such models is known as neural mass models (NMMs), which simulate the dynamics of neuronal populations by providing a balance between mathematical simplicity and biological plausibility. They are currently widely used as generative models to explain noninvasive electrophysiological brain measurements such as those found in EEGs. NMMs describe neural function at a mesoscopic level (Deco et al. 2003; Coombes 2010), in contrast to single neuron ‘integrate-and-fire’ models (Abbott 1999), or the more elaborate Hodgkin and Huxley models (Hindmarsh and Rose 1984; Hodgkin and Huxley 1952). NMMs quantify the mean firing rates and mean postsynaptic potentials of distinct neuronal populations, called neural masses (NMs). While single neurons are considered to be the primary computational units of the brain’s architecture (Hodgkin and Huxley 1952; Finger 2001), it is also widely accepted that the relevant information processing underlying brain function in both healthy and diseased states is carried out by ensembles of interacting neu-
123
Biol Cybern
rons at the mesoscopic level (Shepherd 1991; Shepherd et al. 1975; Jirsa 2004). The first NMMs were proposed in the early 1950s (Beurle 1956). Substantial contributions were made by Freeman and co-workers as part of their study of perceptual processing in the olfactory system (Freeman 1978; Eeckman and Freeman 1991). Freeman underscored that the simplicity and generality of the elements comprising his model make it applicable to other neural systems (Freeman 1987). Similar ideas were developed by Lopes Da Silva et al. and led to the design of a lumped-parameter model, which was able to generate an alpha rhythm (Lopes da Silva et al. 1974, 1976). Starting with the lumped-parameter model, Jansen et al. developed a biologically inspired mathematical framework for simulating spontaneous electrical activities of neuron assemblies as recorded by an EEG, with a particular emphasis on alpha activity (Jansen et al. 1993; Jansen and Rit 1995). The framework has been used to explain epilepsy-like brain activity (Wendling et al. 2000), as well as various narrow-band EEG oscillations, ranging from the delta to the gamma frequency bands (David and Friston 2003). Nevado-Holgado et al. (2012) developed a multi-objective genetic algorithm that can estimate parameters of a neural mass model from clinical EEG recordings. Estimating the parameters of neural mass models generally involves the Bayesian inversion of dynamic causal models using standard variational system identification techniques. Much scientific work has been devoted to the understanding of the behavior of neural mass models. Examples include the derivations of stability conditions of the synchronized state in arbitrary networks (Belykh et al. 2005; Hennig and Schimansky-Geier 2008) as well as the derivations of the equations for the synchronization manifold (DeMonte et al. 2003; Jirsa 2008) for heterogeneous networks. More general, when the system exhibits strong deviations from the synchronized state, mode decomposition techniques become more appropriate (Assisi et al. 2005; Stefanescu and Jirsa 2008). The bifurcation theory of nonlinear systems is a useful tool for investigating the dynamical behavior in neural mass models. For example, Grimbert and Faugeras (2006) used bifurcation analysis to investigate the effect of the input to a NMM on the dynamics of the model and found that rhythmic activities such as alpha rhythm and epileptic wave are related to the structure of a set of periodic orbits and their bifurcation. Touboul et al. (2011) proposed a general framework for studying the bifurcations of neural mass models defined by ordinary differential equations. Breakspear et al. (2005) explained the essential difference between the topology of the global bifurcation diagram of absence and tonic-clonic seizures. NMMs are now used routinely in the dynamic causal modeling of invasive and noninvasive electrophysiological time
123
series (David et al. 2006). In this context, NMMs are used as forward models of hidden neuronal states (that are then mapped to sensors). This provides a complete generative model of observed electrophysiological signals. Dynamic causal modeling (DCM) has been applied to numerous paradigms in the cognitive neurosciences using both EEG, MEG (David et al. 2006; Garrido et al. 2007; Kiebel et al. 2008, 2009), and LFP recordings (Moran et al. 2008). Generally, dynamic causal models use multiple NMMs that are coupled with extrinsic connections. Furthermore, they incorporate explicit delay parameters accounting for within (intrinsic) and between (extrinsic) neural mass connections (Kiebel et al. 2006). In this paper, we focus on a single neural mass with multiple populations and provide a quantitative bifurcation analysis of intrinsic temporal delays. Animal studies suggested the existence of multiple thalamocortical and corticothalamic pathways, yielding a range of thalamocortical delays from 0.1 to 5 ms and longer corticothalamic delays from 1 to 30 ms in the mouse (Salami et al. 2003; Liu et al. 2001), rat (Sawyer et al. 1994; Beierlein and Connors 2002), rabbit (Swadlow and Weyand 1981; Swadlow 2003), cat (Wilson et al. 1976; Tsumoto et al. 1978; Tsumoto and Suda 1980; Miller et al. 2001; Sirota et al. 2005), and macaque monkey (Briggs and Usrey 2007). Scaling the above data to human brain size (Robinson et al. 2004), one can estimate delays of roughly 10 ms for the human thalamocortical pathway and a few tens of milliseconds for the corticothalamic pathway. The work of Roberts and Robinson (2008) indicated that seizure waveforms in the cortex and thalamus depend on the axonal and synaptodendritic delays. Previous work of models of brain has not explicitly taken into consideration delays in connections. Although delays make the analysis more difficult, their effects may be important for the applications. Research on Hopfield-type neural networks with delays first introduced by Marcus and Westervelt (1989) has also shown that delay can modify the global dynamic in interesting ways. Transmission delay seems to play a significant role in integration of information arriving to a single neuron in different spatial and temporal windows and also at the network level in interneuron communication. The theoretical study of the dynamics of simple units organized into networks with delayed couplings revealed a rich variety of possible scenarios of transition to a global oscillatory behavior induced by the delay (Bungay and Campbell 2007; Campbell et al. 2005; Guo 2005, 2007; Song et al. 2005; Yuan and Campbell 2004; Yuan 2007; Wei and Velarde 2004). The emerging oscillations can exhibit different patterns sensitive to the delay. Since then, delays have been inserted into various simple neural networks. Campbell et al. (2004) investigated the stability and bifurcations in the delayed neural network of two coupled three-neuron subnetworks. In this work, we modify the model proposed by Jansen et al. by adding a time delay in the inhibitory feedback loop
Biol Cybern
to simulate signal transmission among neurons, resulting in a neural mass model with time delay. The new model is able to reproduce several different types of EEG activities including alpha wave, interictal EEG, and ictal EEG, with all parameters (excepting extrinsic inputs) being kept constant and equal to standard values. Through an analysis of the influence of the time delay on model’s dynamical behaviors, we succeed in showing that a transmission delay can make the model to produce seizure-like activity.
2 Model description As shown in Fig. 1, the model consists of three subsets of neurons, namely the main cells (i.e., pyramidal cells), the excitatory interneurons, and the inhibitory interneurons. The influence of neighboring areas is represented by an excitatory input p(t) (modeled by band-limited white noise) that globally describes the average density of afferent action potentials. The model output y(t) corresponds to the sum of postsynaptic potentials in activated pyramidal cells, which can in turn be used to synthesize EEG activity. In our model, each subset of neurons involves two operators. The first transforms the average pulse density of action potentials into an average postsynaptic membrane potential, described by the response functions given by Aate−at t ≥ 0 (1) h e (t) = 0 t
