REPORT MODELLING OF DENDRITIC COMPUTATION : COMPARISON OF SUB-LINEAR AND SUPRA-LINEAR INTEGRATIONS FOR DIFFERENT TIME INTERVALS BETWEEN EPSPS

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1 REPORT MODELLING OF DENDRITIC COMPUTATION : COMPARISON OF SUB-LINEAR AND SUPRA-LINEAR INTEGRATIONS FOR DIFFERENT TIME INTERVALS BETWEEN EPSPS Mélodie Durnez Cogmaster, 2nd year Supervisor : Dr. Boris Gutkin Research Director Group for Neural Theory, Laboratoire de Neurosciences Cognitives, Institut d'études de la Cognition, École Normale Supérieure. June 2013

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3 Modelling of Dendritic Computation Context This work has been carried out as part of the internship of the 2nd year of the Cogmaster, from October to January one day a week, and from February to June full time. This internship took place at the Group for Neural Theory, a team of the Laboratoire de Neurosciences Cognitives of the École Normale Supérieure (29 rue d'ulm, Paris). The Dr. Boris Gutkin, one of the Principal Investigators of this team, was my supervisor. Originality declaration Although this work is strongly related to previous studies, the issue and the model are novel. There is no study in which the sub-linear summations and the supra-linear ones are computed from the same model. The way the previous knowledges are used is more described in the main body of this report. Contribution declaration The Dr. Boris Gutkin, my supervisor, proposed me to do a work about dendritic computations, and gave me some bibliography. He advised me for some specific methods for my work, for the interpretation of the outcomes, and he reviewed this report before I submitted it. Besides I got great help from Romain Cazé, a former PhD of the team, who has been working on the same topic as me. He assisted me for defining the issue, developing the global method, and dealing with computer programming. He also commented on each of my results. Finally, I often discussed about physiology and programming tricks with Julien Clauzel, another student in the team who carried out a research parallel to mine. I designed the model and programmed it entirely by myself, I interpreted the figures and wrote the report by myself too. I thought mainly alone about the discussion. 1

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5 Modelling of Dendritic Computation CONTENTS Abstract... 5 A Introduction Framework of the subject Experimental and theoretical background Specific issue Methods B A little bit of physiology Dendrites in the brain A dendrite in details C The set up model Computational material Parameters: Morphology, physiology, and variables D Results Measured versus expected amplitudes Ratios measured to expected amplitudes Areas under curves Involvement of time constants Conclusion E Discussion Justification of the methods Comparison with the outcomes of other studies Further investigations of biophysical mechanisms Implications for the dendritic computations Overall conclusion: openings and limits Acknowledgements References

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7 Modelling of Dendritic Computation ABSTRACT Neurons are units that are able to process information. They receive synaptic inputs, integrate the information they contain, and fire action potentials as outputs. Many experimental and theoretical works demonstrated that the integration of the excitatory post-synaptic potentials, a type of synaptic input, is non-linear. Indeed, when multiple inputs arrive synchronously, the induced modification of the membrane voltage does not exactly match the sum of the individual potentials if they had been triggered separately. Moreover, it is known that this linearity tends to disappear when the time interval between the inputs increases. But the difference between the sub-linearity and the supra-linearity robustnesses to time has not been explored yet. In this study, I built a neuron model that allows to make this comparison. I obtained that the sub-linearities does not exist anymore for time intervals longer than 100 ms, whereas supralinearities still persist after 100 ms. This outcome provides a new rule for the dendritic computations, because it sets down the distinct time windows on which the non-linearities may take place. This adds to the several other mechanisms that can have a part in dendritic integration of inputs and which highly enhance the computational power of the neuron. 5

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9 Modelling of Dendritic Computation A INTRODUCTION 1 Framework of the subject Why did I choose to develop a model of neuronal physiology? Cognitive Science is a field of research whose one of the main goals is to shed light on the links between brain and mind. A lot of disciplines in this area attempt to make this relation, including Neuro-physiology. To my mind, looking at the microscopic level of the brain is a pertinent way to investigate the links between brain and mind, because here are the concrete and detailed bases of cognition and information processing. In Computational Neuroscience, one of the tasks we can achieve is to model neurons and their physiology, by the means of systems of equations. Furthermore, complex computer models and simulations permit to reproduce information processing between and within neurons. However, there are many parameters that can change between neurons, and some types of neurons are totally different. Therefore there are still no fully detailed rules describing the varied computations that a neuron (and a network of neurons) can perform. In this study, we seek to advance a bit further in this research of computational rules. The issue Dendrites are the branches of a neuron from where the informations come in. Then the neuron processes these informations, and finally it fires an action potential through the axon. Therefore this response depends both on the inputs and on the computation made from them. But how is this integration performed? The main paradigm is that the more inputs the neuron received on its dendrites, the greater the probability to produce an action potential. Nevertheless, this proportionality does not always imply a linearity. In other words, there is not always only a multiplying factor between the amount of inputs and the probability of output. Thus, there are sometimes non-linear intergrations. Both experimental and theoretical knowledges support this theory. In my work I investigate some characteristics of these non-linear forms of computation. 7

10 2 Experimental and theoretical background Linear integration of dendritic inputs Neurons fire action potentials according to the inputs they receive from their dendrites. An intuitive rule would be that the more input they get, the more probable is the emission of an action potential. In fact, inputs generate small voltage signals, usually Excitatory Post-Synaptic Potentials (EPSPs) that add up, and sometimes this sum may cross a voltage threshold and then a spike is fired. Indeed, when the whole EPSP is recorded, the value of this potential should be exactly the sum of the individual EPSPs (figure 1). This is genuine proportionality between the inputs and the response. Therefore, if we plot several recorded amplitudes of EPSPs as a function of the corresponding expected amplitudes, (i.e. the artificial sum of the individual EPSPs recorded separetely), we get a straight line that is the bisector of the graph (figure 2). Figure 1: Exemple of linear summation of two identical EPSPs. Figure 2: Measured versus expected amplitude of EPSP. 8

11 Modelling of Dendritic Computation As this dendritic computation can be represented with a linear function, the integration is said «linear». This type of summation is observed in many experiments, for exemple in the work of Cash et al. (1998). Non-linear computations Recent experimental evidences proved that the dendritic integration could be non-linear: the measured EPSP was not equivalent to the expected one. Some authors (Poirazi et al., 2003, Polsky et al., 2004, Muller et al. 2012) recorded EPSPs larger than expected, expressing supra-linear integrations, and some others (Abrahamsson et al., 2012) found out EPSPs lower than expected, that showed sub-linear computations. Some models for these kinds of integration have been developed. They can account for the mechanisms that underlie the supra-linearities (Gomez-Gonzalez et al., 2011) or the sub-linearities (Abrahamsson et al., 2012). For these non-linear integrations, the above plots may become as following. Figure 3: Exemples of non-linear summation (sub-linear in this case). Figure 4: Measured versus expected amplitude of EPSP (sub-linear case). 9

12 Concerning the biological mehcanisms that allow those computations, the sub-linearity is due to a saturation of the accumulated EPSPs received in the dendrite, and the supra-linearity is a consequence of a generation of dendritic spikes, because of a threshold crossing. The physiology of these mechanisms will be further developed in part B. Involvement of time One parameter that can influence the dendritic integration of the EPSPs is the time interval between them. In most studies, it has been observed that the more the inputs were temporally close, the more the non-linearity was pronounced. But no research has been carried out in order to compare the evolution of the supra-linearity with the one of the sub-linearity when varying this time interval. 3 Specific issue The aim of this study is to investigate the evolution of the non-linearities when the time between EPSPs changes. If this time interval increases, the non-linearity will disappear. But is there a difference between the ways the supra-linearity and the sub-linearity die out? Is one progression faster than the other? If so, we would be able to say that one non-linearity is more time-resistant than the other. This would complete the actual knowledges upon dendritic computations, and be a new starting point to investigate the multiplicity of operations that a dendrite can perform. To answer this issue, I made computer simulations with models built on the same bases, in order to be comparable. This had never been done before. 4 Methods This research mainly uses tools of Computational Neuroscience. The model is made with the programming languages NEURON ( and Python ( The first simulation environment computes the basic equations of neurophysiology, that comes from theories of Computational Neuroscience. The second one is a basic 10

13 Modelling of Dendritic Computation programming language, that can be combined with NEURON in order to help managing the simulations and the figures. 11

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15 Modelling of Dendritic Computation B A LITTLE BIT OF PHYSIOLOGY... The aim of this section is to remind the essential biological features about dendrites and their processing of information, before to describe the model and to present the outcomes. 1 Dendrites in the brain The function of the brain is to process information. Like computers, it receives inputs and shapes outputs. The 1010 neurons in the brain are important cells, because they compose the network that perceives external or internal stimuli, processes information, and emits commands to muscles. But each neuron itself is able to do this processing. Indeed, each neuron is made up with three main parts: the dendritic arbor, composed of the branches that receives inputs from other neurons, the soma, that is the cell body that gathers the signals from dendrites, and the axon, which sends a response toward other neurons. dendrites soma axon Figure 5: Structure of neuron, exemple of a pyramidal cell. Sizes of neurons can vary widely, therefore the soma can be between 5 and 120 μm large. The diameter of the axon is between 1 and 15 μm, and its length can be from 1 mm to more than 1 m long. The dendrites have a decreasing diameter from the soma to the distal end (about 8 to 0.2 μm), and their length can be between 15 μm and 2 mm. 13

16 The connexions between neurons are the synapses. Each neuron displays an average of 104 synapses on its dendrites. When the axon of the pre-synaptic neuron emits an action potential, the tip of the axon releases neurotransmitters, that are collected by the receptors on the post-synaptic dendrite. This receiving is a specific signal, that is tranformed by the means of molecular mechanisms, and results in a change of the cell membrane voltage. The several inputs received by the dendrites will then be integrated, the neuron handles the information, and a spike is made in response to the input. Figure 6: Schema of a synapse. An hypothesis can be made, that is the more developed dendritic tree the neuron has, the more complex computations it will be able to perform. This theory is confirmed in the study of Gollo et al. (2009). 2 A dendrite in details Receiving of the input Different kinds of neurotransmitters can be released by the pre-synaptic ending. There exist dozens of types of neurotransmitters, but only one type is produced by a neuron. In this study, the most represented neurotransmitter is used: the glutamate. It is an excitatory neurotransmitter, that 14

17 Modelling of Dendritic Computation means that releasing it will excite the post-synaptic neuron, tending to make it active and to fire an action potential. Each neurotransmitter can be bound to specific receptors, because of their molecular configuration. The receptors for glutamate that will be used here are the α-amino-3-hydroxy-5méthylisoazol-4-propionate (AMPA) and the N-méthyl-D-aspartate (NDMA) receptors. When binding a molecule of glutamate, these two receptors will induce an EPSP, but by different means. Membrane potential and initiation of the EPSP In the body, there are positive and negative ions, present in different concentrations inside and outside the cells. If these ions were free to move inward or outward the cell, they would arrange according to two constraints: a chemical one and an electrical one. Indeed, if there is a too large concentration of the same molecules, or a too large amount of the same polarities, on one side of the cell membrane, some ions will tend to cross the membrane until the installation of an equilibrium. This is the electrochemical equilibrium. In neurons, instead of this passive diffusion, there are «ionic pumps» that force some ions ot go in or out of the cell. These active mechanisms make the system going away from the equilibrium, and maintains a gradient of potential. Consequently to these active spread mechanisms, there are more potassium (K +) and proteins (mainly negative) inside the neuron, and more sodium (Na +), chloride (Cl-), and calcium (Ca2+) outside. Their respective concentrations result in a polarisation of the membrane: there are more negative charge inside than outside. Thus, at rest, the neuron membrane displays a negative voltage, typically -70 mv. extracellular K+ proteins- membrane intracellular Na+ Ca²+ Cl- Figure 7: Schema of intracellular and extracellular concentrations. 15

18 If one type of ions is suddenly enabled to freely cross the membrane, it will move in order to recreate an electrochemical equilibrium. Consequently, the membrane voltage will evolve in the direction of the equilibrium potential (or «reversal potential») that is specific to the ion. This specific equilibrium potential is described by the Nernst equation: E= RT Cin ln zf Cout E is the equilibrium potential (in mv), R is the gas constant (in J/(mol*K)), T in the temperature (in K), z is the charge of the ion, F is the Faraday constant (in C/mol), and Cin and Cout are the intracellular and extracellular concentrations of the ion (in mol). The difference between the equilibruim potential of the ion and the current voltage (EionV(t)), because it will pull the potential toward another value, is called the driving force. Let us go back to the neurotransmitter receptors. The AMPA receptor is a ionic channel, a gate on the post-synaptic cell membrane, that can let or not pass sodium and potassium ions. When it binds a neurotransmitter, the gate opens (by chemical means). Sodium comes inside the cell and potassium goes outside. But the balance is an overall entry of positive charges, thus the membrane voltage increases towards positive values: it is a depolarisation. This changing of potential is an Excitatory Post-Synaptic Potential. As regards the NMDA receptor, it is a ionic channel too, but it needs first another event: the removal of a ion of magnesium bound on it, which prevents the specific ions from passing through the channel. This action can be done only if there is a small post-synaptic depolarisation: the removal is voltage-dependent. This preceding depolarisation may be achieved by the means of a previous activation of an AMPA receptor. Consequently, the openning of this channel takes more time than for the AMPA receptor, and needs sufficiently previous input to depolarise enough the membrane. Then, cations enter into the cell (mainly calcium, and some sodium), and others go out of it (potassium). The balance is, as for AMPA receptors, a depolarisation of the neuron membrane. 16

19 Modelling of Dendritic Computation Figure 8: AMPA and NMDA receptors. Another temporal aspect in the initiation of an EPSP, lies in the time the receptors take to bind and unbind the molecule of glutamate, and the time of its own working. These are time constants, which are characteristic of each receptor. Propagation of the EPSP along the dendrites Once the EPSP is initiated by the synaptic receptor, the depolarisation will propagate along the dendrite until the soma. This can be achieved passively or not. A dendrite has the same properties as an electrical cable: a diameter, a length, a conductance and thus an intrinsic resistance. Therefore, the depolarisation can porpagate passively. However it is noteworthy that there is a loss of intensity when the distance travelled increases, because of the existence of leaks, that can be due to open channels or branchings for exemple. Besides, there are active properties that can influence the EPSP propagation. Along the dendrite, there are voltage-dependent channels: they open or close as soon as the membrane voltage moves beyond a threshold. NMDA receptors, because of their voltage-dependence for the removal of the ion magnesium, can contribute to this active propagation, if they are located on the EPSP pathway. The distinction between passive and active properties will be usefull for understanding the upcoming results. 17

20 All EPSPs, because of the conductivity of the cell, propagate toward the soma. They may be attenuated, suppressed, or enhanced on their pathway, but once at the soma they can trigger or not an action potential, which will move forward through the axon. Integration of multiple EPSPs EPSPs can sometimes be superimposed, for several reasons: because they are close in time, close in space, because they meet at a dendritic node, or when they arrive together at the soma. As seen in the introduction, these voltages can sum linearly or not. If they sum linearly, the measured amplitude will be exactly the sum of the amplitudes of the two individual EPSPs. But they may sum non-linearly, and then the measured amplitude can be larger or smaller than the expected one. In general, sub-linearities are due to the passive properties of the dendrite: when two excitatory inputs arrive sufficiently close in time and space, the first of them will induce a depolarisation, and when the second one arrives it will be less strong than expected because its driving force is reduced: the membrane voltage already started to increase, thus the difference between the actual voltage and the equilibrium potential of the ion related to the EPSP is reduced (Rall et al, 1967, London et al., 2005). When a dendrite displays this sub-lineartity, one can say that it saturates. Supra-linearities may be due to active mechanisms: voltage-gated channels and NMDA receptors can enhance rapidly the EPSP amplitudes. Because this mechanism is like the one that triggers action potentials in the axon, the enlarged EPSP is called a dendritic spike. Indeed, the EPSP increases the membrane potential, that opens voltage-dependent channels, which in turn let pass ions that extends again the depolarisation (Polsky et al., 2009). The summation of multiple EPSPs can, by these manners, be sub-linear or supra-linear. Those kinds of integration have been already highlighted experimentally and theoretically, as said in the introduction. 18

21 Modelling of Dendritic Computation To keep in mind Biochemical and biophysical mechanisms present in the dendrite allow it to perform different computations, that enables the neuron to process complex informations. Passive features implies a sub-linear summation of the EPSPs, owing to a reduction of the driving force. Voltage-gated channels and NMDA receptors, on the other hand, permit a supra-linear summation, because they can enlarge the size of the EPSP, and even trigger dendritic spikes. 19

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23 Modelling of Dendritic Computation C THE SET UP MODEL We just saw that EPSPs can interact with each other, if they are enough close in time. Some authors indeed noticed that the larger is the time interval between inputs, the less pronounced is the non-linearity, and the summation tends to become linear for largest intervals (Polsky et al., 2004, Polsky et al., 2009, Abrahamsson et al., 2012, Gomez-Gonzalez et al., 2011). The aim of this study is to compare how the sub-linearity and the supralinearity are robust when time intervals are extended. For this pupose, I created a computational model of a neuron, in order to perform simulations of dendritic inputs with several time intervals, and record the produced EPSPs. The advantage of this method, unlike the experimental one, is to allow to control well the parameters, and the possibility to investigate easily the links between the observed effect and the inserted mechanisms. 1 Computational material The model is implemented with two programming environments: NEURON and Python. Each of these environments has a specific language. Both are commonly used in scientific studies: the first one is the most employed to reproduce artificial neurons with detailed physiology, and the second one is a basic play script which permits to intuitively implement the desired algorithms. In this work, I use both environments at the same time, in the following manner. I write my instructions in a Python sheet, and I import the NEURON software at the beginning of this script. Thus I write the model mainly with Pyhton language, but using specific terms that call for NEURON variables, objects, or equations. Therefore the biophysical features are performed through NEURON system, and the simulation running is made with Python play script. Then these instructions are executed via Python. 21

24 2 Parameters: Morphology, physiology, and variables Modelling of the neuron and the inputs The model of neuron is the simplest one, in order to be the more possible representative of several types of neurons, and to avoid bias. It includes one soma and one dendrite, and one synapse and the distal end of the dendrite. Recording Stimulation 0 DENDRITE SOMA 1 SYNAPSE Figure 9: Drawing of the neuron used in the model. In the script, both dendrite and soma have their ends labelled «0» and «1». The «1» end of the dendrite is connected to the «0» end of the soma. The parameters for these compartments are the same as in the model of Abrahamsson et al. (2012), for the neuron to be able to display sublinearity. I chose their model and not someone's else because it is relevant for sublinear summation in dendrites, it is simple, and all parameters are available in the paper: For the soma: diameter = 9 μm. axial resistance = 150 Ω.cm. It is the intern resistance of the dendrite, in the length direction of the dendrite. membrane capacitance = 0.9 μf/cm². It is the ability of the membrane to store electrical charges on its both sides. 22 passive mechanisms are inserted passive conductance = S/cm². It is the ability of the membrane to let ions cross it.

25 Modelling of Dendritic Computation For the dendrite, parameters are the same except: diameter = 0.4 μm. In both sections, all other parameters have the values by default of NEURON, among others: global membrane reversal potential = -70 mv. The neuron reproduced in Abrahamsson et al. (2012) is a cerebellar interneuron, but these values used can fit a great variety of neurons. The synapse is at the «0» end of the dendrite. Two different types of synapses are used, to allow either sub-linearity or supra-linearity. An «Exp2Syn» synapse is used for the sub-linear summation of EPSPs. It is an object available in the NEURON library, which stands for a synapse with AMPA receptors. On the other hand, the synapse used to induce supra-linearities is an NMDA synapse, not available in the NEURON platform, but that can be found on a database of models written to work under NEURON. This model of synapse has been created by Gasparini et al., as part of their study whose results were published in I chose this model of NMDA synapse because Romain Cazé (the former PhD of the team who has also been working on non-linearities in dendrites) advised it to me, because it is built according to our subject and works well for our issues. Synaptic currents, the movements of ions through the postsynaptic membrane when an input arrives, are set to 40 na. A «network connector» is set on the synapse. It is an artificial element used in NEURON for the synapse to be able to receive a stimulation. The EPSP is produced by a «network stimulation», which is a NEURON tool that can induce multiple successive stimuli, with defined time intervals. Two inputs are given to the tip of the dendrite, the time interval between them varies across simulations. For the supra-linearity, when this time equals 0, it is not possible to trigger the two pulses at the same time on the same network connector (because of the operating of the synapse), so I created two networks connectors on the same synapse, and induced one pulse on each. The simulation lasts 150 msec, and the resting voltage is set to -70mV. The first input is induced 1 msec after the beginning of the simulation. All others parameters are the parameters by defaut of NEURON. 23

26 Controlled and measured variables Across simulations, some parameters are varied in order to address the issue. First, the time interval between the stimulations, which is in milliseconds. The second parameter that is changed is the strength of the synapse. This number indicates how the post-synaptic neuron is receptive to the input, how much it will respond to the stimulation. This value may stand, for exemple, for the quantity of receptors on the post-synaptic membrane: the more there are receptors, the more the input is amplified. This parameter is varied in order to get several sizes of EPSPs, across which the linearity of the sum of the EPSPs will be investigated. The measured variable is the membrane voltage at the middle of the soma (located at «0.5» in the model), which would display the EPSP. Two features of this EPSP are computed: its maximum amplitude, and the area under its curve. The first one will serve to investigate the non-linearities in the amplitudes summations. The measured amplitude is recorded during a simulation, and the expected one is computed using two other independent simulations. In theses two latest simulations, only one input is delivered on each neuron, at moments corresponding to the times of the inputs for the first simulation. The two obtained EPSPs are then summed, and the maximum amplitude is taken down. The area of the EPSP is measured from the discrete points computed by the program, that are multiplied by the time step of the run (0.025 ms, default value of NEURON), and finally added up. The expected and measured areas are computed as above for the maximum amplitudes, in order to make a comparable graph. Some other numbers are calculated: the measured to expected amplitude ratio, and the measured to expected area ratio. This will allow to estimate the amount of non-linearity: if the number is lower, equal, or higher than one, we would be able to interpret if there is a sub-linearity, a linearity, or a supra-linearity respectively. This is a better way to precisely characterise the nonlinearity, than only looking at the measured versus expected graph, because it will permit to quantify the amounts of sub-linearity and supra-linearity, and to compare them by symmetry. 24

27 Modelling of Dendritic Computation D RESULTS The issue is to compare how the sub-linearity and the supra-linearity evolve when the time interval between two dendritic inputs is varied. To address this, I begin by computing and plotting the expected and measured amplitudes of the EPSPs, in order to observe the global evolution of the non-linearities for several time intervals between the stimuli. 1 Measured versus expected amplitudes Sub-linear summation of EPSPs I first sought to display sub-linear summations of EPSPs, starting from the parameters used in Abrahamsson et al. (2012), with passive conductances and an AMPA synapse. I got the following result: Figure 10: Measured amplitudes of EPSPs versus the expected ones, for several time intervals between stimulations. See the curve for simultaneous EPSPs, that is where the time interval is 0 ms. For lowest expected amplitudes, the measured amplitude is the same as the expected one. Thus the sum is linear. But when the expected amplitude increases (because of a reinforcement of the strength of the synapse), the curves goes under the graph bisector, the measured amplitude of the EPSP is lower 25

28 than the expected one. As the expected amplitude increases, the curve is further and further from the bisector, so the sum is more and more sublinear. Therefore the model reproduces well the nonlinearity observed in the litterature, for dendrites with passive mechanisms. When the time interval between the inputs increases, the curve approaches the bisector. The summation is less and less sub-linear as the duration between the EPSPs increases. This outcome could have been intuited a priori, because the sublinear summation is due to the first EPSPs that induces a reduction in the driving force for the second; consequently when the duration between the inputs increases, the membrane has more time to repolarise, and the driving force becomes again large enough to allow linearity. The summation becomes totally linear for time intervals from 100 ms. This value seems then to be the upper time boundary for sub-linearities, for neurons with the parameters implemented in my model. Supra-linear summation of EPSPs Afterwards I looked to obtain supra-linearity. I kept the same model, for my outcomes to be comparable, but I only changed the synapse. I put an NMDA synapse instead of an AMPA one, which would lead to an amplification of the signal, as we saw above. Figure 11: Measured amplitudes of EPSPs versus the expected ones, for several time intervals between stimulations. As before, if we look at the curve plotted for the 0 ms time interval, we can see that it is 26

29 Modelling of Dendritic Computation quite far from the bisector, but over it. At the beginning, for low amplitudes, the summation is rather linear, but for an expected amplitude of about 4 mv, there is a sharp increase in the non-linearity. Suddenly, the measured amplitude of the EPSP is much bigger than the expected one. There seems to be a threshold, after which the EPSP is very higher than it should be. For this reason, it can be called a dendritic spike, a feature I mentionned in the part about dendritic physiology. This is therefore supra-linearity. When the expected amplitudes increases more after the threshold, the measured amplitude increases no more so roughly, it has reached a king of plateau. The amplitude of the dendritic spike does not increase very much. There is a sort of saturation, because of which, from about 17.5 mv, the summation becomes linear and immediately after it turns sub-linear, as for the AMPA synapse. This is due to the inability of the dendrite to trigger EPSPs larger than this plateau, when the stimulations are to close, probably because of the lack of driving force. When the time interval increases, as for sub-linearity, the curve goes nearer the bisector, thus the supra-linearity decreases. Because of the temporal distancing of the EPSPs, they interact less with each other. When the time advances, more and more NMDA channels activated by the first stimulus are deactivated, thus they are no more available to trigger the spike when the second EPSP arrives. As regards the final sub-linearity, the sum becomes more linear too when the time interval increases, for the same reasons as for the global sub-linearity observed in the previous paragraph. For a time interval of 100 ms, the sum is still not fully linear across the expected amplitudes, thus the time window on which supra-linearity is possible is a little longer than 100 ms. 2 Ratios measured to expected amplitudes We want to quantify the amount of non-linearity, and then to look at its evolution when the time interval increases. In order to have an estimate of the quantity of non-linearity, we need a number that can indicate how far is the measured versus expected curve from the bisector. We thus have to compare the values of the plotted curve with the matching values of the bisector, for each expected amplitude. We can do the difference between these values, or their ratio. Here we choose the ratio in order to cancel the influence of the order of magnitude of the EPSP amplitude, that could be a bias. 27

30 Plots of the ratios as a function of time intervals For each expected amplitude, we can compute the amount of non-linearity, by doing the ratio of the measured amplitude to the expected one. We want to investigate the course of this ratio for several time intervals between EPSPs. For this goal I plotted the ratios versus the time intervals for a given expected amplitude, and then superimposed the plots for the different amplitudes. Figure 12: Ratio measured to expected amplitude of EPSP, versus the time intervals between stimulations, for several expected amplitudes. Here is the result for the model of sub-linearity. When the ratio equals 1, the measured amplitude equals the expected one, that means that the sum is linear. When the ratio is lower than 1, the measured amplitude is lower than the expected one, that stands for a sub-linearity. As we saw in the figure 10, we can see here that whatever the expected amplitude, the sum becomes linear when the time interval increases. Moreover, the sublinearity is greater for larger amplitudes of EPSPs. Besides, we can see that even for an amplitude of 1 mv there is sub-linearity for short intervals, that we could not notice on the previous graph. Concerning the supra-linearity, we got the following plot. Most values of the ratios are higher than 1, because the measured amplitudes are greater than the expected ones. For amplitudes lower than 17 mv, the supra-linearity decreases as the time interval increases. Increasing the expected amplitudes between 1 mv and 5 mv, the supra-linearity rises, and between 5 mv and 17 mv it falls down, because of the plateau corredponding to the dendritic spike. For expected amplitudes higher than 17 mv, the sum is sub-linear, because of the saturation of the dendrite. For 28

31 Modelling of Dendritic Computation the main supra-linearity, as for the saturation, the sum becomes more linear as the time interval between the inputs increases. Figure 13: Ratios measured to expected amplitude of EPSP, versus the time intervals between stimulations, for several expected amplitudes. These graphs are somehow redundant with the previous ones, since they bring little new information, but they allow a better viewing of the evolution of the non-linearities when increasing the duration between the EPSPs. In addition, this will permit to compare the evolutions of the sublinearity and the supra-linearity when increasing the time intervals. Comparison of the evolutions of the non-linearities Here is the core of the study, because the outcomes form the response to the issue. I took some comparable curves from the two previous figures, and I plotted them on the same graph, in order to compare their evolution when time intervals increase. First, I chose the curves for which there is the same amount of non-linearity when the EPSPs are simultaneous. In this way, the curves have symetrical intercepts regarding the value 1, and their evolutions can be compared. I took two curves for each non-linearity, with measured to expected ratios of 1 ± 0.09 and 1 ± 0.28 when the time interval equals 0. The corresponding expected amplitudes are 3 mv and 15 mv concerning the sub-linearity, and 1 mv and 13mV for the supralinearity. All sums become linear when the time interval increases, but this occur faster for the neuron with sub-linearity than for the neuron with supra-linearity. 29

32 Figure 14: Superimposition of selected curves from the two previous figures, in order to compare the evolutions of the non-linearities versus time intervals. Therefore the computations supported by supra-linear mechanisms will be more timeresistant than those involving sub-linear mechanisms. The implications of these observations are futher developed in the discussion. Then I chose to plot other curves, which would be also comparable a priori: the ones computed for the same expected amplitudes. I picked out the curves with the lower, the higher, and an intermediate expected amplitude (1 mv, 19 mv, and 7 mv respectively). Figure 15: Superimposition of selected curves from the figures 12 and 13, in order to compare the evolutions of the non-linearities versus time intervals, for the same expected amplitudes. 30

33 Modelling of Dendritic Computation For same expected amplitude, the amounts of non-linearity are very different. Both the intercept and the evolution with increased time intervals are different. For exemple, the supra-linear sum is much more pronounced than the sub-linear one for an expected amplitude of 7 mv. We can see well the dendritic spikes here. Besides, if we look at the curves for the expected amplitude of 1 mv, we can see again that the way the non-linearity disappears when increasing time intervals seems faster for the neurons that display sub-linearity. 3 Areas under curves The maximum amplitude of an EPSP may be not enough to characterise it. Indeed, its width may also vary. Both measures may be related with each other, but there is no automatic relation between them. That is why we propose here to take an interest in the areas under EPSPs, which once combined with the maximum amplitude, can offer an idea of the width of the EPSPs. To investigate the evolution of the area under curve when changing the time interval between EPSPs, I performed the same analyses as before, and compared the outcomes. First, I produced the measured versus expected graph for the area under curve, concerning the sub-linearity. Figure 16: Measured versus expexted amplitude, and measured versus expected integrals of EPSPs. Here we can observe that the curves have the same shapes. If the curves for the area showed 31

34 linearity, that would have meant that the area were conserved while the amplitude diminished, in this case the measured EPSP would have been spread, compared with the expected one. Here, the amplitude is modified in the same way as the whole EPSP. When the measured amplitude equals the expected one divided by a number x, the area is also divided by this number x. This proves that the EPSP, if having a regular shape, is reduced in its globality, and not flattened and widened. As regards the supra-linearity, the maximum amplitude and the area under the curve have the same overall behavior when increasing time intervals: the supra-linearity (for lower amplitudes) and the saturation (for greater amplitudes) tend to vanish. However the curves are rather different if compared with each other. Figure 17: Measured versus expected amplitude, and measured versus expected integrals of EPSPs. The conversion from supra-linear summation to the saturation occurs sooner for the more little time intervals, and later for the longest ones. In the first case, this means that there is a point from which the measured peak EPSP is bigger than expected, whereas the integral is smaller. Therefore, the measured EPSP would be narrower than expected. In the second case, for great EPSPs and long time intervals, it is the inverse, the amplitude is summed sub-linearly, whereas the area is summed almost linearly. Therefore, the shape of the EPSP would be more flatter and wider comparing with the expected one. This result shows that the supra-linear summation of EPSPs is not just an overall enlargement of the linear sum. The shape of the EPSP is also modified. This modification is a new aspect, that could have a role to play in the computations that a dendrite can perform. In order to 32

35 Modelling of Dendritic Computation investigate this transformation of the shape, it would be better to use experimental data, that provide more precise and realistic plots of EPSPs than the data from this model, because a lot of physiological mechanisms are not included here. It is not an appropriate method to study the details of a simplified model in order to find out subtle features. Then I plotted again the graphs concerning the ratios, but from the areas under the EPSPs instead of the maximum amplitudes. Figure 18: Ratios measured to expected amplitude of EPSP, and ratios measured to expected areas. For the sub-linearity (figure 18), the curves are almost the same. This supports the fact that the sub-linear summations are essentially global over the EPSP. But if we pay attention to the leftside scale, we can see that for time intervals lower than 40 ms, the sub-linearity is more marked for the areas than for the amplitudes. The EPSPs should then be narrower than expected, in a stronger way than the amplitude reduction is decreased. If we turn to the plots for the supra-linearity (figure 19), we can find again the features discussed for the measured versus expected plot. For the greatest EPSPs, for the smallest time intervals there are more sub-linearities in the area than in the amplitude of the EPSPs, and for longest time intervals, there are less sub-linearities in the area. This confirms the interpretation of the figure

36 Figure 19: Ratios measured to expected amplitude of EPSP, and ratios measured to expected areas. Now we compare the evolutions of the non-linearities for the areas under the EPSPs. As before, I plotted some selected curves from the previous figures (figure 20). Figure 20: Superimposition of selected curves from the two previous figures, in order to compare the evolutions of the non-linearities versus time intervals. I chose the curves, for sub-linearity and supra-linearity, that have the same intercept. We can see the same difference in their evolutions as when regarding the EPSP amplitudes: a faster linearisation of the sub-linear sums when increasing time intervals, than the supra-linear sums. The left and right curves are not exactly the same, but it would not be really pertinent to compare their differences in detail, because they are not obtained with the same synaptic strengths, 34

37 Modelling of Dendritic Computation thus they do not come from the same EPSPs. In conclusion, because both amplitudes and areas under the EPSPs seem to have the same general evolution when increasing time intervals, we can write out that the extinction of both nonlinearities occurs across the whole EPSP. Not only their amplitude, but also the area under the curve, undergo the non-linear transformation and the disappearance of this non-linearity. Furthermore, the sub-linearity attenuates faster than the supra-linearity, for both amplitudes and areas of EPSPs. Thus, it is the whole sub-linear EPSP that is faster «linearised» than the supralinear one. Therefore, the EPSP in its globality is more or less sensitive to the time interval between the inputs. This outcome is an indication concerning the biophysical mechanisms that underlie the non-linear integration of dendritic inputs. Next I again compare curves from the figures 18 and 19, but I chose the ones that are plotted for the same EPSP amplitude or area values. Figure 21: Superimposition of some curves from the figures 18 and 19, in order to compare the evolutions of the non-linearities versus time, for the same expected amplitudes or areas. As for the amplitudes, the graph concerning the areas displays very different curves for the sub-linearity and for the supra-linearity, for the same expected value. As regards the smallest EPSPs (amplitude of 1 mv or area of 100 mv.ms), the sub-linearity vanishes faster than the supra-linearity, for both EPSP amplitude and area. For the two other sizes of EPSP, for both EPSP amplitude and area, the curves for the sub-linear and supra-linear cases are very different, they are not symmetrical in relation to the horizontal «ratio = 1» line. 35

38 Overall, however, when increasing time intervals, both non-linear summations of areas seem to become linear more quickly than for the non-linear summations of the amplitudes. This means that, when the time duration between the inputs increases, the width of the EPSPs approaches faster the one expected, than the amplitude does. The exact physiological mechanisms underlying those features would need to be experimentally investigated, but we may already assume an involvement of the time contants specific to the synaptic receptors. These time constants could indeed have a huge role to play in the evolution of the non-linearities when the time interval between the EPSPs is varied. 4 Involvement of time constants Definition of time constants and possible effects on dendritic integration Time constants can be two kinds of parameters. First, it may affect the voltage of the EPSP, in this case time constants are some durations that are specific to an EPSP: typically its rise time and its decay time, that is to say the times spent to reach precise levels of voltage. Second, it may concern the membrane ionic channels. Here the time constants describe timings of the opening and the closing of a channel, when the appropriate signal (a neurotransmitter, a specific voltage, etc.) is present to activate it. During an EPSP, time constants of the voltage greatly depend on the several time constants of the involved channels. In our study, the non-linearity of the sum of EPSPs comes from the interaction between these EPSPs. If the EPSPs are too temporally distant from each other, they do not superimpose and the sum is linear. But if they overlap, the location of this overlapping within the time courses of the EPSPs will possibly influence the voltage computation. We have observed this effect for the evolution of the non-linearities when the time interval between EPSPs varies. Taking into account the remarks about time constants, we can notice that, in addition to the duration between the inputs, the time constants of the EPSPs themselves will take part in determining the overlapping of the EPSPs: for a same time interval, large EPSPs will be more likely to superimpose than small ones. That is why the shapes of the EPSPs need to be investigated here. 36

39 Modelling of Dendritic Computation Investigation of EPSPs time courses In this part, we study the global shapes of the EPSPs, in order to find whether their time course can explain the results observed concerning the non-linearities. Indeed, if we come across very narrow EPSPs for the sub-linearities, and wider ones for the supra-linearities, this would explain why the supra-linear summations are more robust than the sub-linear ones when the time interval between inputs increases: it would be due to the fact that the EPSPs still overlap whereas the sub-linear ones have become distinct, for greatest time intervals. Besides, in the simulations, some parameters are varied: the time interval between the inputs, and the synaptic strength. Thus, if we want to investigate the shapes of the EPSPs, we have to take into account both variations, because they can modify the voltage course. That is why, for each non-linearity, I plotted several situations, that combine different synaptic strengths and time intervals. The first figure concerns the sub-linearity, and the second one affects supra-linearity. In each figure, EPSPs are shown for time intervals of 0 ms, 40 ms, and 80 ms (one column for each duration), and chosen synaptic weigths (marked «w» on the plots) of 0.005, 0.01, and 0.1 (one row for each strength). Expected EPSPs are in blue, measured ones in green. Figure 22: EPSPs obtained in simulations allowing sub-linear summations. 37

40 Figure 23: EPSPs obtained in simulations allowing supra-linear summations. In figure 22, we can observe sub-linear summations whatever the synaptic weight: the green curve is always underneath the blue one. In figure 23, there are supra-linear summations for synaptic strengths of and 0.01, but a sub-linearity for a synaptic weight of 0.1. This is in line with the plots of measured versus expected amplitudes: for largest EPSPs, the dendrite saturates. However, I chose to plot the EPSPs corresponding to this range of weights because they display well the interesting features. When changing the synaptic weight, we can easily see here that the width of the EPSPs varies. Indeed, when increasing the synaptic strengths, the widths of the EPSPs increase too, and grow faster in the supra-linear simulations (figure 23) than in the sub-linear ones (figure 22). Consequently, if we take, for exemple, a time interval of 0 ms, there is only one weight for which both sub-linear and supra-linear EPSPs have about the same width: the value For higher or lower values, the widths of sub-linear and supra-linear EPSPs are not equivalent. We were looking for visible time constants of the EPSPs, that would have been strikingly different for the sub-linear and the supra-linear cases. Indeed, if the supra-linear EPSPs had been generally larger than sub-linear ones, that would have induced interactions between supra-linear EPSPs even for longest time intervals, whereas the sub-linear ones would have been purely separated. Here we observe that these time constants are not fixed, but strongly depend on synaptic 38

41 Modelling of Dendritic Computation strengths. Therefore, the fact that supra-linear summations are more time-resistant than sub-linear ones (which was the conclusion of the previous outcomes), does not seem to be a result of distinct time constants between the non-linearities. Thus, what may be the cause of the robustness of the supra-linear summations? If it is not about time constants or overlappings of EPSPs, it might be due to deeper physiological mechanisms, like precise behaviors of synaptic receptors. These hypotheses need to be further investigated, by the means of both electrophysiology and modelling. 5 Conclusion Summary of the outcomes Firstly, I used my model to reproduce the well-known figures of measured versus expected maximal amplitudes of EPSPs, that allow to display the sub-linear or supra-linear summations occurring in the dendrites. I added variations across the time intervals between the inputs, in order to observe the evolution of these non-linearities. I could notice that both non-linearities vanish when the time interval increases. Then I computed the ratios of measured to expected amplitudes, in order to better characterise the amounts of non-linearity. This permitted me to compare the evolutions of the nonlinerarities when the time interval between EPSPs varies. We noticed that the supra-linear summations were more robust than sub-linear ones, when increasing the durations. Thirdly, we studied the areas under the EPSP curves, instead of the maximal amplitudes. This allowed to investigate if it was the sum of the whole EPSP or only its amplitude that was nonlinear, and to follow if this persisted in the same manner as the maximum amplitude when increasing time intervals. We found that, in general, the areas under the curves behaved on the same way as the maximal amplitudes, throughout all analyses made on them. Finally, the study of the observable widths of the EPSPs indicated that there are no strict differences between sub-linear and supra-linear EPSPs that could have accounted for the larger robustness of the supra-linear summations when the time interval between inputs increases. Gathering of the data, possible interpretation 39

42 If we gather all these outcomes, we can say that the non-linear summations concern the whole EPSPs, not only their amplitudes. Besides, the sub-linear summations attenuated faster than the supra-linear ones when the time interval increases. This is not explained by a difference in the witdhs of EPSPs, but probably by distinct physiological mechanisms involved for both nonlinearities. Advance within the theory The non-linearities observed in dendritic integrations allow the neuron to perform a very wider range of possible operations than if there were only linearities (Cazé et al., 2013, Gollo 2009). Moreover, the time interval between the inputs may affect greatly these computations (Agmon-Snir et al., 1998, about coincidence detection, and Polsky, 2009). Here, we found a skecth of rule for the involvement of the time interval in the non-linear summations of EPSPs: the distinct robustnesses of the sub-linear and the supra-linear computations in relation to the durations between the inputs. Therefore, the computations permitted by these two kinds of summations will not have the same efficiency when produced within the same range of time intervals. Consequently, when one wants to study the multiplicity of computations that dendrites may perform, one will have to take into account that the non-linearities are more or less efficient across these specific ranges of time intervals. 40

43 Modelling of Dendritic Computation E DISCUSSION This section is a critical view of the study. Furthermore it positions the outcomes within the current theoretical and experimental knowledges, and it proposes directions for future examinations. 1 Justification of the methods The reasons for which the issue has been adressed through a computational method, and the justification of the way the model has been developed were mentioned in section C. However some aspects of these choices might be questionned. First, the simplicity of the model. One could argue that there is not enough details to permit an application of the rules extracted here to dendrites in general. Actually, the aim here was to get a general idea on the distinct time windows on which the non-linearities occur, and to compare them on the basis of similar models. Moreover, the features observed here could be a starting point to further characterise the time-resistances of the non-linerarities for more specific types of neurons. Second, the fact to use the same model for both non-linearities could also be called into question, since these non-linearities involve, by definition, different physiological mechanisms. In addition, in most experiments, the types of neurons that display one of the two kinds of nonlinearity are totally different. Nevertheless, here, the outcomes plots prove that the same neuron can display both sub-linear and supra-linear integrations, only by changing the synapse. Therefore, this simple model is in fact a good basis to investigate distinct properties of the non-linearities. The third point, to my mind, requires more attention. It is a technical one, that affects the use of the ratios to determine the amount of non-linearity. Let us take an exemple: if the measured amplitude of the EPSP is two times the expected one, the obtained value for the ratio is 2. Besides, if the measured amplitude in two times less than the expected one, the value of the ratio is 0.5. Thus, these points will not be symetrical in relation to the horizontal «ratio = 1» line, while a same factor is used to compute the measured EPSPs. This asks the question of the operations made in the non-linear computations: is the EPSP multiplied or divided, or is there a certain quantity of voltage that is added or subtracted from the expected EPSP? In the first case, the use of the ratios might not be a suitable method to compare the non-linearities, for the reason just described in the example. 41

44 There, the right indicator would be the measured to expected ratio for the supra-linearity, and the expected to measured ratio for the sub-linearity, in order to extract the multiplying factor. In this study it is the second case that is assumed, and the normalisation to the expected amplitude is an appropriate method in this framework. 2 Comparison with the outcomes of other studies The main outcome is counter-intuitive, given previous results When we addressed the issue, we thought that the sub-linearity would be more time-resistant than the supra-linearity. Indeed, in most studies, the supra-linear summation needs a very narrow temporal window, to allow a great overlapping of the EPSPs, so as to get enough depolarisation to cross the voltage threshold and trigger a dendritic spike. For this reason, the dendrite that displays supra-linearity is often associated to a «coincidence detector», as in Agmon-Snir et al. (1998). The sub-linearity, in the other hand, did not present such restricted time range, and besides it relies on passive mechanisms, that are usually slower than the active ones. Consequently, we believed that the time window for the supra-linearity would have been shorter than the one for the sub-linearity. Here we obtained the opposite. This proves that every intuition, even strong, needs to be checked. Moreover, this asks new questions about the distinct modes of functioning of the non-linearities. Expected versus measured plots As said in the introduction, many authors pointed out non-linear summations with plots of the measured versus expected amplitude of the EPSP. The figures I obtained match well the ones of Polsky et al. (2004) and Abrahamsson et al. (2012). But the comparison with the result of GomezGonzalez et al. (2011) is less staightforward, partly because they did not use the same scale. Indeed, they focused on the low amplitudes of EPSPs (below 6 mv), whereas I got amplitudes until 20 mv. Thus, in the figure 24-left, we cannot see the supra-linearity for the time intervals of 2 ms, 3 ms, and 5 ms, because they probably occur for greater amplitudes, that are out of the plot. As regards the curve for a time interval of 0.1 ms, the observable supra-linearity may be due to their use of a very detailed model, with many different currents included inside. 42

45 Modelling of Dendritic Computation Figure 24: Measured versus expected amplitudesof EPSP, from Gomez-Gonzalez et al. (2011) (left), and from my model (right). The measured versus expected plots of Polsky et al. (2004) and Abrahamsson et al. (2012), that matched better my data, come from experimental recordings, whereas the plot of GomezGonzalez et al. (2011) is from a model. Thus we could assume that the genuine range of EPSPs amplitudes is from 0 mv to 20 mv. However, Magee et al. (2000), found that a single synaptic event would trigger an EPSP whose amplitude is between 0.8 mv and 1 mv. Therefore, higher measured amplitudes may result from multiple synaptic events due to the artificial stimulation, and in my model the high amplitude may result from the large synaptic strengths I used. Influence of time intervals on the non-linearities Some authors already investigated how the non-linear summations evolved when varying time intervals between EPSPs, even if they did not compare the evolution of both non-linearities. Here I compare their results with mine. First, Polsky et al. (2004), produced two stimulations on a dendrite of a pyramidal cell, which were distant from 20 μm. They varied the time interval between the stimulations, and recorded supra-linear EPSPs. They computed the expected amplitudes of EPSPs by summing individual EPSPs. Then they plotted the expected and measured amplitudes on the same graph, as a function of time intervals (figure 25). They found that the dendritic integrations begin to be linear for time intervals from 50 ms. There could be two reasons for the difference with my finding of 43

46 more than 100 ms. On one hand, it could be because of the spatial distance between their stimulations, that could permit an attenuation of the EPSP, by the means of passive mehcanisms. On the other hand, they have only one measured EPSP for each time interval, whereas I got more because I varied the synaptic strengths. Thus, their data are only a sample of mine, and if we attempt to replace their points on my figure, we can observe that they nearly coincide with some points of my plots. Figure 25: Measured versus expected amplitudes of EPSP, from Polsky et al. (2004) (left), and from my model (right). Second, we can compare my results with those of Abrahamsson et al. (2012). They also used a ratio to characterise the amount of non-linearity, computed in the same manner as our indicator, but then transformed into a percentage. They plotted this ratio versus the several time intervals, but varying the distances between the stimulations instead of the synaptic strengths. Figure 26: Amounts of non-linearity, from Abrahamsson et al. (2012) (left), and from my model (right). 44

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