Propiedad Intelectual en el Ecuador

In Chapters 4 and 5 I use a number of advanced statistical techniques, which will be detailed here. Unless otherwise stated I made pairwise comparisons by performing Mann-Whitney’s U test at the 5% significance level, and means are quoted ± the standard error of the mean. To control for the familywise error rate I applied the Bonferroni correction for multiple comparisons.

In Chapter 5 I use principal component analysis (PCA) to investigate the sources of variability in the dataset, along with Gaussian mixture models, linear discriminant analysis, and random forests in an attempt to automatically discriminate between cell classes. Prior to performing any of these techniques I transformed log-normally distributed parameters into log-space, so that they were normally distributed, to remove any bias towards high numerical values of a given parameter. After this I normalised the dataset by calculating theZ-score:

Z = (µi−X)/σi, (3.20)

whereµi and σi are the mean and standard deviation, respectively, of parameter i.

Since random decision forests, introduced by Breiman (2001), are a non-standard technique in neuroscience I shall describe the method of fitting and using them here in detail. A random forest is constructed of many individual classification trees that are each trained on a bootstrapped sample of the dataset. During classification each tree in the forest classifies the object independently, or ‘votes’ for the class. The forest output is then the class with the most votes. For a dataset of N objects (i.e. cells), each with M parameters (i.e. measured cell parameters). Each tree is then grown as follows:

1. A bootstrapped sample of size N is taken from the original dataset. This sample is then used for growing the tree, and the remaining objects are referred to as out-of-bag (OOB).

2. A number m < M is chosen. At each node m parameters of the total M are chosen at random and the best binary split (i.e. each node as two child nodes) is decided upon based on these. This m value is held constant over the entire forest.

3. Each tree is grow in full with no pruning.

The forest classification error rate depends on two things: the correlation between the trees, an increase in which increases the forest error rate; and the strength of each tree (i.e. how good a predictor they are), an increase in which decreases the forest error rate. Using a large m value increases both of these and a small m

decreases them, so there is an optimum value ofm somewhere in the middle. When training the tree, the best split at each node is that which minimises the impurity at the two resulting child nodes. Impurity can be defined in several ways, but here I use Gini’s diversity index. This is defined by

G= 1− c

X

i=1

f_i2, (3.21)

where cis the number of classes (in this case four) and fi is the fraction of objects

at the node of class i; if all objects are of the same class then G = 0, otherwise

G > 0. Minimising G minimises the node’s impurity. The best split is then that which minimises Gat each of the child nodes.

There are two additional features that make random decision forests appealing for classification. Firstly, they provide an unbiased estimator of the classification error rate without the need for a separate test set or performing cross-validation. To calculate this, consider thekth tree in the forest. After thekth tree has been trained the OOB objects are the classified using that tree. The jth _{object in the dataset is}

OOB in about one third of the trees (Breiman 2001). The proportion of times that thejthobject is not classified as its true class is averaged over all objects, giving an estimate of the forest error rate. This error rate decreases roughly monotonically as the number of trees in the forest increases before reaching a plateau. At this point additional trees will not provide any more predictive power.

The second appealing feature of random decision forests is that they provide an estimate of the importance of each parameter used for classification. To calculate the importance of parameteri, the algorithm proceeds as follows:

1. For each tree in the forest, classify the OOB objects and count the number of correct classifications.

2. Randomly permute the values of parameter i within the set of OOB objects for each tree.

3. Re-classify the OOB objects with permuted parameters and again count the number of correct classifications for each tree.

5. To obtain the parameter importance score of parameter icalculate the stan- dard error of the result of step 4 from all trees in the forest.

To examine correlations in the dataset I calculated both Spearman’s rank cor- relation coefficient and the standard covariance. To calculate Spearman’s rank cor- relation coefficient, ρ, for a sample sizen, the raw parameter values Xi and Yi are

first converted into ranksxi and yi, respectively, after which ρ is defined by

ρ= 1−6

Pn

i=1d2i

n(n2₋₁₎, (3.22)

where di = xi −yi. All data and statistical analyses was performed with custom

Chapter 4

Physiological Quantification of

Neocortical Pyramidal Cells

4.1

Introduction

Pyramidal neurons are the principal excitatory cell in the neocortex and display het- erogeneity in their morphology (Oberlaenderet al. 2012; van Aerde and Feldmeyer 2013; Laramee et al. 2013; Marx and Feldmeyer 2013), electrophysiology (Nowak

et al.2003; Zaitsevet al.2012; van Aerde and Feldmeyer 2013; Marx and Feldmeyer 2013), synaptic dynamics (Wanget al.2006), and projection targets (Thomson and Lamy 2007). Specifically, quantifying the electrophysiology of different cell types helps us understand how neurons perform computations and relay information to the rest of the network.

Cell physiology is typically described by the cells’ responses to a series of depo- larising and hyperpolarising current steps. Whilst these inputs are simple and easy to interpret, they are not necessarily representative of those received in vivo. Fur- thermore, quantifying the physiological heterogeneity using these methods is often not useful for constructing a meaningful model. To address this one can quantify electrophysiology by fitting a model to a cell’s response to a naturalistic stimuli that is representative of in vivo-like activity. Producing models that capture enough biological realism yet are simple enough to analyse, along with designing efficient algorithms with which to fit them is currently the focus of much research in compu-

tational neuroscience (Gerstner and Naud 2009). The complexity of these models ranges from the continuous biophysically realistic that capture the full action poten- tial dynamics and can include dozens of ionic channels (Section 1.3.1), to the simple one-dimensional discontinuous that are primarily concerned with the precise timing of action potentials (Section 1.3.2).

Although attempts have been made to fit detailed continuous models (Huys et al. 2006; Druckmann et al. 2007; Huys and Paninski 2009; Hay et al. 2011), this is hugely challenging task due to their high dimensionality potentially leading to non-identifiability. Discontinuous integrate-and-fire models have proven to be far more popular due to their low dimensionality and mathematical tractability. Fur- thermore, it has been suggested that they are more relevant for somatic recordings than single-compartment, isopotential continuous models due to their sharp spike initiation (Brette 2013). Numerous discontinuous models and fitting methods have been suggested with varying degrees of success (Jolivet et al. 2008). Typically, the models that best fit both the sub-threshold response and spike timings have a dy- namic threshold (Badelet al.2008a,b; Kobayashiet al.2009; Yamauchiet al.2011), spike-triggered adaptation current (Brette and Gerstner 2005; Clopathet al. 2007), or both (Mensi et al. 2012). In practice, combining naturalistic stimuli with more traditional approaches will provide the most complete description of a cell’s electro- physiology.

In this chapter I quantify the somatic electrophysiology of somatosensory cortical layer 2/3, layer 4, and slender- and thick-tufted layer 5 pyramidal cells. I extract key parameters from the cells’ responses to a combination of square-pulse and nat- uralistic in vivo-like stimuli during whole-cell patch-clamp recordings. Many of the parameters were extracted from the cells’ dynamicI-V curve (Badelet al.2008a,b), a consequence of which is the generation of reduced neuron models that accurately replicate the experimental voltage time-course (a MATLAB toolbox to implement this analysis is provided in Appendix A). All cell classes studied were found to fit the exponential integrate-and-fire (EIF, Fourcaud-Trocm´e et al. 2003) form with class-dependent parameter statistics. I quantify the fit quality of the dynamic I-

quantity rather than optimising directly to the voltage response, it is comparable or superior to that of alternative fitting approaches: the spike response model (Mensi

et al. 2012), the multi-timescale adaptive threshold model (Kobayashiet al. 2009), and the adaptive exponential integrate-and-fire model (Brette and Gerstner 2005). I go on to investigate between-class differences in the post-spike response of the model parameters, with particular focus on the dynamics of the resting potential. Finally, I determine significant differences between cell classes and fit marginal parameter distributions.