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Transport processes across the membrane are essential to ensure survival of every living cell. Therefore, the exchange of membrane impermeable molecules is mediated by specific transport proteins, which are embedded in the lipid bilayer.
One important class comprises secondary active transporters, which couple very efficiently the uphill transport of the main substrate against its concentration gradient to the downhill transport of an additional substrate. These transporters are widely distributed among all kingdoms of life and accomplish many crucial functions. One function is to counteract the deleterious effect of hyperosmotic stress in bacteria. Several members of the BCCT (betaine-choline-carnitinetransport) family of secondary transporters mediate osmostress protection by the accumulation of the compatible solute betaine or its precursor choline (Lamark et al., 1991; Peter et al., 1996; Ziegler et al., 2010). Besides osmo-dependent sodium or proton-coupled symporters, the BCCT family includes few rare representatives of osmo-independent transporters such as the substrate:product antiporter CaiT from E. coli (Jung et al., 2002; Ziegler et al., 2010).
The best-characterized member of the BCCT family is the sodium-coupled betaine transporter BetP from Corynebacterium glutamicum. BetP together with the ABCtransporter OpuA and the H+-solute symporter ProP, became a paradigm for osmoregulated osmolyte transport. Although, all three transporters were extensively studied, the general mechanism of osmoregulation is still far from being understood. Thus, one task of this thesis was to elucidate further the regulatory properties of BetP.
BetP is tightly regulated by osmotic stress and is able to increase its basal betaine uptake activity dramatically upon elevated osmolalities within one second (Peter et al., 1998a). The osmotic stress is sensed by BetP via two stimuli, one is the increase of the internal K+ concentration above a threshold of 220 mM (Rübenhagen et al., 2001), the second is related to a change in the physical state of the membrane (Maximov et al., 2014). So far, several solved crystal structures in combination with functional and computational analysis provided insights into the coupling mechanism of betaine and its co-substrate sodium (Khafizov et al., 2012; Perez et al., 2012). Despite the wealth of data, the precise regulatory mechanism of trimeric BetP is still unclear.
Different approaches are possible when it comes to modeling the brain. Given its biological nature, models can be constructed out of the chemical and biological building blocks known to be at play in the brain, formulating a given mechanism in terms of the basic interactions underlying it. On the other hand, the functions of the brain can be described in a more general or macroscopic way, in terms of desirable goals. This goals may include reducing metabolic costs, being stable or robust, or being efficient in computational terms. Synaptic plasticity, that is, the study of how the connections between neurons evolve in time, is no exception to this. In the following work we formulate (and study the properties of) synaptic plasticity models, employing two complementary approaches: a top-down approach, deriving a learning rule from a guiding principle for rate-encoding neurons, and a bottom-up approach, where a simple yet biophysical rule for time-dependent plasticity is constructed.
We begin this thesis with a general overview, in Chapter 1, of the properties of neurons and their connections, clarifying notations and the jargon of the field. These will be our building blocks and will also determine the constrains we need to respect when formulating our models. We will discuss the present challenges of computational neuroscience, as well as the role of physicists in this line of research.
In Chapters 2 and 3, we develop and study a local online Hebbian self-limiting synaptic plasticity rule, employing the mentioned top-down approach. Firstly, in Chapter 2 we formulate the stationarity principle of statistical learning, in terms of the Fisher information of the output probability distribution with respect to the synaptic weights. To ensure that the learning rules are formulated in terms of information locally available to a synapse, we employ the local synapse extension to the one dimensional Fisher information. Once the objective function has been defined, we derive an online synaptic plasticity rule via stochastic gradient descent.
In order to test the computational capabilities of a neuron evolving according to this rule (combined with a preexisting intrinsic plasticity rule), we perform a series of numerical experiments, training the neuron with different input distributions.
We observe that, for input distributions closely resembling a multivariate normal distribution, the neuron robustly selects the first principal component of the distribution, showing otherwise a strong preference for directions of large negative excess kurtosis.
In Chapter 3 we study the robustness of the learning rule derived in Chapter 2 with respect to variations in the neural model’s transfer function. In particular, we find an equivalent cubic form of the rule which, given its functional simplicity, permits to analytically compute the attractors (stationary solutions) of the learning procedure, as a function of the statistical moments of the input distribution. In this way, we manage to explain the numerical findings of Chapter 2 analytically, and formulate a prediction: if the neuron is selective to non-Gaussian input directions, it should be suitable for applications to independent component analysis. We close this section by showing how indeed, a neuron operating under these rules can learn the independent components in the non-linear bars problem.
A simple biophysical model for time-dependent plasticity (STDP) is developed in Chapter 4. The model is formulated in terms of two decaying traces present in the synapse, namely the fraction of activated NMDA receptors and the calcium concentration, which serve as clocks, measuring the time of pre- and postsynaptic spikes. While constructed in terms of the key biological elements thought to be involved in the process, we have kept the functional dependencies of the variables as simple as possible to allow for analytic tractability. Despite its simplicity, the model is able to reproduce several experimental results, including the typical pairwise STDP curve and triplet results, in both hippocampal culture and layer 2/3 cortical neurons. Thanks to the model’s functional simplicity, we are able to compute these results analytically, establishing a direct and transparent connection between the model’s internal parameters and the qualitative features of the results.
Finally, in order to make a connection to synaptic plasticity for rate encoding neural models, we train the synapse with Poisson uncorrelated pre- and postsynaptic spike trains and compute the expected synaptic weight change as a function of the frequencies of these spike trains. Interestingly, a Hebbian (in the rate encoding sense of the word) BCM-like behavior is recovered in this setup for hippocampal neurons, while dominating depression seems unavoidable for parameter configurations reproducing experimentally observed triplet nonlinearities in layer 2/3 cortical neurons. Potentiation can however be recovered in these neurons when correlations between pre- and postsynaptic spikes are present. We end this chapter by discussing the relation to existing experimental results, leaving open questions and predictions for future experiments.
A set of summary cards of the models employed, together with listings of the relevant variables and parameters, are presented at the end of the thesis, for easier access and permanent reference for the reader.