Time series of events play an important role in neuroscience and are often described by point processes on the real line. A first and far-reaching example is information processing in the brain. By integration of input from adjoining cells or from sensory organs such as the ears and eyes, nerve cells (also called neurons) transmit electric signals – so-called spikes – to adjacent cells so that information is processed. In precise terms, spikes are short periods (of length 1-2ms) of a typical increase in the neuron’s membrane potential. As the duration and the height of this increase do not differ remarkably, it is commonly assumed that the information content of neuronal activity is mainly coded by the temporal sequence of spiking the so-called neuronalspike train. For the understanding of the nervous system, it is therefore crucial to develop a deep insight into neural firing activity. A detailed explanation of the role of neurons and the whole nervous system can be found in, e.g., the textbooks of Kandel et al. (2000) and Berg et al. (2007).
Formally, a spike train is given by the sequence (t1, t2, . . . , tn) of spike times in a recording interval [0,T] (with 0< t1 < t2 < . . . < tn< T <∞). The intervals between consecutive spike times are called inter-spike intervals or life times. Spike trains with usually hundreds of events are a well studied object in computational neuroscience, where statistical models based on point processes like renewal processes are used frequently (Johnson, 1996; Dayan and Abbott, 2005; Kass et al., 2005; Nawrot et al., 2008; Gr¨un and Rotter, 2010).
Often stationarity of the process parameters like rate or variance of the life times is required for further analyses such as coordination between parallel point processes (e.g., Gr¨un and Rotter, 2010). Change points in the rate may cause misinterpretations of serial correlations when assuming a constant rate (Farkhooi et al., 2009). The impact of neglecting non-constant parameters on techniques assuming stationarity is further discussed in, e.g., Brody (1999);
Gr¨un et al. (2003). Hence, it is crucial to capture potential change points, and non-stationary spike trains are in a preprocessing step often split up into sections with approximately constant parameters (Schneider, 2008; Staude et al., 2010; Quiroga-Lombard et al., 2013). To detect changes in the rate, considerable research has been conducted, e.g., by Fryzlewicz (2014);
Messer et al. (2014); Eichinger and Kirch (2018), compare also the reviews of Khodadadi and Asgharian (2008); Aue and Horv´ath (2013); Jandhyala et al. (2013). Little is known about the detection of variance change points, especially with rate change points being present. To the best of our knowledge, the only theoretical work dealing with variance changes in presence of
1. Introduction and outline
a non constant mean is Dette et al. (2015). An example of a spike train with both – rate and variance changes – is given in Figure 1.1. The occurrence times of the spikes are symbolized by the vertical bars.
Figure 1.1: A point process with a non-stationary rate and variance profile.
Detected changes of the rate or the variance not only improve statistical analysis by separating stationary periods but might also contain important information themselves. Different firing patterns as described, e.g., in Bingmer et al. (2011) are connected to changes in variability.
For instance, in dopamine neurons the firing patterns often switch between a low-rate regular or irregular single spike background pattern and short so-called ”bursty” periods with a large number of spikes. These bursty periods represent a possible change in variance and have been shown to be coupled to an increase in dopamine release (e.g., Gonon, 1988; Schiemann et al., 2012).
A second example showing the application of time series of events in neuroscience areresponse patterns to behavioral experiments like ambiguous stimuli. The perception of ambiguous stimuli changes spontaneously in an unpredictable and subjective manner. Traditional examples of ambiguous stimuli are the Necker cube (Necker, 1832) or Rubin’s vase (Rubin, 1915), see Figure 1.2. In these examples, there are two possible perceptions such that we also speak of bistable perception.
Figure 1.2: Necker Cube (A) and Rubin’s vase (B). In the Necker Cube either the upper-right or the lower-left square can be interpreted as front side. Rubin’s vase may be perceived as vase or as two faces looking at each other. The graph is slightly modified from https://commons.wikimedia.org/wiki/File:Multistability.svg (Public Domain license).
Recently, also rotating spheres with switching perceived rotation direction were used in bistable perception experiments, e.g., Schmack et al. (2013, 2015). The increasing sequence of perceptual reversal points (t1, t2, . . . , tn) in a recording interval [0, T] is called response pattern, and the periods of constant perception are calleddominance times. Examples for continuous and intermittent presentation (i.e., with short blank displays between the presentation phases of the stimulus) are shown in Figure 1.3.
We observe an increase in variability from continuous (green) to intermittent presentation (blue) in Figure 1.3 compared to the respective mean dominance times. During continuous
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1. Introduction and outline
presentation the dominance times appear to be unimodal distributed, whereas during in-termittent stimulation phases of rapidly changing perception interchange with long stable phases (reported also by Brascamp et al., 2009). Modeling these different types of response patterns as well as linking them to possible underlying neuronal mechanisms in a model with only a few interpretable parameters such that also possible differences between groups can be explained is a challenging task. Current approaches often use detailed assumptions and large parameter sets, which complicate parameter estimation.
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Figure 1.3: Examples of response patterns to a bistable stimulus. Response pat-terns to continuous (green, A,B) and intermittent (blue, C-F) presentation from the data set reported in Schmack et al. (2015). Each of the six response patterns shows the responses of one individual to continuous presentation (recording time 240s, green) or intermittent presentation (recording time 1200s, blue) of the bistable stimulus. While the distribution of dominance times tends to be unimodal in the continuous case, stable and unstable phases seem to interchange in intermittent stimulation. In addition, response patterns can be highly variable across subjects.
Main goals of the thesis
The overall goal of the thesis is to describe and detect different kinds of variability changes in point processes on the real line analyzed in neuroscience as introduced above. The thesis is divided into two parts. In the first part, we focus on the detection of non-stationarities in the rate and variance (of the life times) of point processes like neuronal spike trains. The main goal is to extend the multiple filter method proposed by Messer et al. (2014) to detect changes in the variance in multiple time scales in presence of rate change points where we assume the rate to be a step function. The method uses a non-parametric approach that is applicable to a wide range of inter spike (or life time) distributions if there are enough events.
The subject of inquiry in the second part are the aforementioned response patterns to bistable stimuli, where in particular data of Schmack et al. (2013, 2015) are used. The main goal of this part is to develop a model that builds a bridge between empirical data analysis and mechanistic modeling and that captures the change in variability from continuous to intermittent presentation. Thus, the model should be able to describe both the response patterns to continuous presentation (with a one-peaked distribution of dominance times) and the response patterns to intermittent presentation where the distribution of dominance times is
1. Introduction and outline
rather bimodal (compare Figure 1.3). Moreover, the model should be fittable to typically short experimental data such that statistical investigation of differences between clinical groups is possible, and the model should allow for neuronal correlates.
In summary, variability changes in point processes should be described in both parts of the thesis, where in the first part a broadly applicable method for change point detection in the rate and the variance is presented, and in the second part we focus on a direct modeling approach with a model enabling links to neuronal processes and describing the variability change in response patterns from continuous to intermittent presentation of a bistable stimulus.