CASTILLO TEMPLARIO

Accurate estimation of the functional connectivity and graph theoretical measures discussed above require epochs of EEG data on the order of seconds in length (Gudmundsson et al., 2007; Fraschini et al., 2016), and assumes stationarity of

Quantifying neural dynamics

the brain dynamics over this epoch. In reality, it is believed the resting state of the brain is composed of a number of ‘resting state networks’ each correspond- ing to different cognitive domains (Lehmann et al.,1998;Britz et al.,2010;Michel and Koenig,2018), and that these networks remain stable for a tens to hundreds of milliseconds (allowing for changes in polarity) before rapidly transitioning to another network (Koenig et al., 1999; Khanna et al., 2015; Michel and Koenig,

2018). EEG microstate analysis is a method used to study this switching be- haviour of the resting state (Khanna et al., 2015; Michel and Koenig, 2018) by studying the instantaneous topographic maps of the EEG.

The first microstate analyses used a method of adaptive segmentation in which the maxima and minima of consecutive EEG topographic maps were com- pared; a microstate was defined as the epoch for which the local maxima and minima stayed within a given window (Lehmann et al.,1987). Wackermann et al.

(1993) followed this procedure and then performed a post-hoc clustering of the centroid locations of each microstate to find that almost all segments belonged to 2-6 (mean 3.7) microstate classes. This information that brain microstates formed a small number of classes lead to the development of k-means clustering meth- ods (Koenig et al.,1999), which not only account for the maxima and minima of the field but the entire spatial topography. Furthermore, these methods allow for definition of regularly occuring maps. A large number of studies using different numbers of electrodes, participants, and filter settings have identified the same four canonical maps in the resting state EEG (see Figure 3 ofMichel and Koenig

(2018) for a review). These four classes have been labelled A, B, C, and D in the literature, and have been correlated with resting state networks related to vari- ous cognitive domains (Britz et al.,2010). These maps have also been validated using other clustering methods such as topographic atomize and agglomerate hierarchical clustering (TAAHC) (Khanna et al.,2014).

Microstates are known to change in healthy development and aging (Koenig et al., 2002) and neurological disorders including dementias (Ihl et al., 1993;

Dierks et al., 1997; Strik et al., 1997; Stevens and Kircher, 1998; Nishida et al.,

2013), schizophrenia (Koenig et al., 1999; Lehmann et al., 2005; Nishida et al.,

2013), and depression (Strik et al., 1995; Atluri et al., 2018). The majority of microstate studies in AD were performed prior to the development of k-means type methods (Ihl et al.,1993;Dierks et al., 1997;Strik et al.,1997;Stevens and Kircher, 1998), meaning alterations to specific classes have not been well char- acterised. Furthermore the differences between temporal scales of microstates and spectral/functional network measures make EEG microstates a prime can- didate for additional information as an electrophysiological biomarker. Therefore understanding changes to EEG microstates in AD and using these alterations as a biomarker is the focus ofchapter 5.

Chapter 2

Modelling single cell dynamics in

tauopathy

This chapter is based on the work published the Journal of Theoretical Biology as

Tait et al.(2018) in collaboration with Dr Marc Goodfellow (supervision, method- ological design), Dr Jon Brown (supervision, conceptualization), Dr Kyle Wedg- wood (conceptualization, methodological design), and Prof Krasimira Tsaneva- Atanasova (supervision, conceptualization). The author’s contribution to this chap- ter includes development and formal analysis of the model, visualization of the results, and writing of the chapter.

2.1

Introduction

The entorhinal cortex occupies a key role in the cortical-hippocampal circuit, act- ing as a gateway between the neocortex and hippocampus (Canto et al., 2008) and playing a pivotal role in working memory processing and spatial navigation (McNaughton et al.,2006;Moser et al.,2008). Many different functional cell types involved in the coding of spatial representation are found in the entorhinal cortex, including grid cells, border cells, head direction cells and speed cells (Hafting et al.,2005;Solstad et al.,2008;Giocomo et al.,2014; Kropff et al.,2015). Spa- tial information from these cells is transferred from layer II of the entorhinal cortex to place cells in the hippocampus, which in turn feed back into the entorhinal cortex (O’Keefe et al.,1998;Deng et al.,2010;Barak et al.,3015).

The principle neurons in layer II of the medial entorhinal cortex are reported to be predominantly (60-70%) stellate cells (mEC-SCs) (Alonso and Klink,1993;

Booth et al., 2016a). Analysis of recordings of mEC-SCs in brain slices demon- strates a number of key identifying electrophysiological properties, including a large membrane potential sag mediated by a hyperpolarisation activated cation current (Ih), subthreshold oscillations in the theta (4-12 Hz) range and clustered action potential firing (Alonso and Klink, 1993). Dorsoventral gradients in these

electrophysiological properties (Giocomo et al., 2007; Garden et al., 2008; Gio- como and Hasselmo,2008,2009;Dodson et al.,2011;Booth et al.,2016a) reflect similar dorsoventral gradients in grid cell spacing (Hafting et al., 2005), implying a key role in spatial memory.

The disruption of memory systems is one of the hallmarks of dementia (Mc- Gowan et al., 2006). The most common cause of dementia, Alzheimer’s dis- ease, has been shown to affect the entorhinal cortex early in disease progression (Braak and Braak (1991); Figure 1.3). Experimental models of tau pathology have suggested a relationship between neurofibrillary tangles and spatial mem- ory deficits (Fu et al., 2017) that may be underpinned by alterations in the intrin- sic cellular dynamics described above (Booth et al., 2016a; Fu et al., 2017). It is therefore crucial if we wish to develop treatments and therapies to build our understanding of the mechanisms underlying mEC-SC dynamics so that we can further elucidate the cellular and network bases of spatial memory, and ultimately the causes and consequences of Alzheimer’s disease.

There are many potential dynamical frameworks within which to mathemati- cally model clustered firing of neurons or the generation of subthreshold oscilla- tions (sections 1.4-1.5). Phenomenological models have used extrinsic rhythmic inputs to drive integrate-and-fire type neurons across bifurcations (Pastoll et al.,

2013; Solanka et al., 2015), thus producing temporal periods of quiescence in- terspersed with bursts of action potentials, that may be reminiscent of clustered firing. Low dimensional neuronal models such as the Izhikevic neuron (which is a non-linear integrate-and-fire type neuron) have been used to model mEC-SC firing patterns (Izhikevich,2007;Shay et al.,2016) but are also constructed from a phenomenological dynamical systems perspective and do not offer mechanistic insight at the single neuron level. For example, they do not allow understanding of the relationship between properties of membrane channels and the aforemen- tioned dynamic firing patterns.

In order to develop a mechanistic, biophysical understanding, Frans ´en et al.

(2004) developed a detailed, compartmental model of an mEC-SC, based on the Hodgkin-Huxley formulation (section 1.5). In addition to standard Hodgkin- Huxley ion channels, hyperpolarisation-activated, cation non-selective channels (Ih) were incorporated along with calcium-gated potassium channels including a potassium-mediated after-hyperpolarisation (AHP) current. It was demonstrated that this combination of channels was sufficient to describe limit cycle subthresh- old oscillations in the theta (4-12 Hz) range and clustered action potential firing. A simulation study of the noise driven system demonstrated a dependence of clustered firing on the AHP conductance and the time scale of the slow Ih com- ponent (Frans ´en et al.,2004). To investigate the role that stochastic effects could play in generating stellate cell dynamics, Dudman and Nolan (2009) formulated

Introduction

a high dimensional, Markov chain model of stochastic ion channel gating and demonstrated that this model could reproduce the aforementioned dynamics due to intrinsic ion channel noise. Clustered action potential firing was generated by a transient increase in probability of action potential firing during recovery from the AHP. This required the Ih current, since simulations and experimental investiga- tion of an Ihknockout resulted in loss of clustering.

These models have provided insight into the potential biophysical mechanisms underpinning the clustered action potential firing and subthreshold oscillations of mEC-SC. However, the dynamic mechanisms underpinning clustered action po- tential firing were not elucidated, which precludes a thorough understanding of the ways in which changes in parameters affect dynamics. Such understanding would help to build a more complete picture of the reasons why different firing patterns can emerge, for example due to diseases such as Alzheimer’s disease. Furthermore, previous models have been cumbersome, either due to their de- pendence on calcium gated-channels or stochastic simulations. A simpler model would allow us to extend more readily into neuronal networks in the future in or- der to better understand the spatial structures underpinning memory processing in health and disease.

In order to advance such a framework, in this chapter, the model of Dudman and Nolan (2009) is converted to the deterministic Hodgkin-Huxley formulation. This results in an ordinary differential equation (ODE) model that retains the key components of Ih and IAHP. As a single compartment model with only voltage- gated ion channels, this model is simpler than the multi-compartment model of

Frans ´en et al. (2004) which includes both voltage- and calcium-gated ion chan- nels. Upon introducing extrinsic noise to the membrane potential in a stochastic differential equation (SDE) framework, numerical simulations are used to demon- strate that this model is capable of generating clustered action potential firing as well as subthreshold membrane potential fluctuations with peak power in the theta band, in line with experimental results. Numerical bifurcation analyses demon- strate that clustered firing in the model arises due to a flip bifurcation (Chan- nell et al. (2007); Barrio and Shilnikov (2011); section 1.4.6). Clustered action potential firing can, in turn, be understood in terms of a fast-slow system (sec- tion 1.4.5), in which the activation of the persistent sodium (NaP) and inactivation of the slow A-type potassium (Kas) channels act as slow variables, driving the fast sub-system through a hysteresis loop via subcritical Hopf and homoclinic bifurca- tions. Thus, in terms of the underlying dynamics, this model can be classified as a subcritical Hopf/homoclinic burster (Izhikevich,2000). This model allows for clus- tered action potential firing to be controlled, making it a suitable model to study the role of dorsoventral gradients in clustering. It is thereby proposed that alterations to AHP or Ih conductances could mediate the quantitative changes in clustering

observed experimentally. In experimental models of dementia (rTg4510), loss of clustered firing is found to correlate with significant changes to AHP amplitude (Booth et al.(2016a);Figure 2.1) but no change in Ih mediated sag (Booth et al.,

2016a). Hence our results suggest a possible path through parameter space that account for the differences in patterned firing in rTg4510.