Investigations of the neural firing threshold

Investigations of the Neural Firing Threshold Carsten Erdmann September 2011 Im Fachbereich Biologie, Chemie, Pharmazie der Freien Universität Berlin eingereichte Dissertation 1. Gutachter: Prof. R. Menzel, Freie Universität Berlin 2. Gutachter: Prof. U. Heinemann, Charité Berlin Tag der Prüfung: 17.4.2012 Hiermit versichere ich, die vorliegende Dissertation selbstständig angefertigt und keine auÿer den angegebenen Hilfsmitteln verwendet zu haben. Ottersberg, 30.9.2011 ................................................. (Carsten Erdmann) Auch die längste Reise beginnt mit einem Schritt. Chinese saying Papa, das geht doch nicht, dass Du jedes Wochenende das ganze Wochenende nur an Deiner blöden Arbeit schreibst! Lena Erdmann Contents 1 Introduction 6 1.1 Approach ............................. 7 1.2 Concepts.............................. 8 1.2.1 2-Dimensional State Space . . . . . . . . . . . . . . . . 8 1.2.2 The Spike Initiation Point (SIP) . . . . . . . . . . . . . 8 1.2.3 The Threshold Separatrix . . . . . . . . . . . . . . . . 10 1.3 Questions to the System . . . . . . . . . . . . . . . . . . . . . 12 2 Precise SIP Detection 13 2.1 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Finding the SIP . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Checking the SIP nder . . . . . . . . . . . . . . . . . 18 2.2 Results............................... 18 2.2.1 SIPFinder......................... 18 2.2.2 Late Spike Phenomenon . . . . . . . . . . . . . . . . . 21 2.3 Discussion............................. 23 2.3.1 Point of Spike Initiation . . . . . . . . . . . . . . . . . 23 2.3.2 Speed of Spike Dynamics . . . . . . . . . . . . . . . . . 24 2.3.3 Late Spike Phenomenon . . . . . . . . . . . . . . . . . 24 3 Neurons have a 2D Firing Threshold 26 3.1 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 Objective ......................... 26 3.1.2 Cells............................ 26 3.1.2.1 Animals..................... 26 3.1.2.2 Slice Preparation . . . . . . . . . . . . . . . . 27 3.1.2.3 Recording Electrodes . . . . . . . . . . . . . . 28 3.1.2.4 The Experimental Setup . . . . . . . . . . . . 28 3.1.2.5 Cell Search and Stimulus Parametrization . . 29 3.1.2.6 Stimulation . . . . . . . . . . . . . . . . . . . 30 3.1.2.7 In Vitro Experiments . . . . . . . . . . . . . 31 1 CONTENTS 2 3.1.3 Models........................... 32 3.1.3.1 Modied Leaky Integrate-And-Fire Model . . 32 3.1.3.2 Hodgkin-Huxley-Model . . . . . . . . . . . . . 35 3.1.3.3 Wang-Buzsáki-Model . . . . . . . . . . . . . . 35 3.1.3.4 Awiszus-Model . . . . . . . . . . . . . . . . . 35 3.1.3.5 Reduced Awiszus-Model . . . . . . . . . . . . 36 3.1.3.6 Modied Awiszus-Models . . . . . . . . . . . 36 3.1.3.7 In Silico Experiments . . . . . . . . . . . . . 36 3.1.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.4.1 Cell Quality Criteria . . . . . . . . . . . . . . 37 3.1.4.2 Data Postprocessing . . . . . . . . . . . . . . 37 3.1.4.3 Data Analysis . . . . . . . . . . . . . . . . . . 37 3.2 Results............................... 38 3.2.1 Cells............................ 38 3.2.1.1 Separatrices . . . . . . . . . . . . . . . . . . . 38 3.2.1.2 Oset ...................... 41 3.2.1.3 Stability and Sensitivity . . . . . . . . . . . . 41 3.2.1.4 Pharmacology . . . . . . . . . . . . . . . . . . 44 3.2.2 Models........................... 44 3.2.2.1 Correctness of Separatrix Construction . . . . 44 3.2.2.2 Separatrices . . . . . . . . . . . . . . . . . . . 44 3.2.2.3 Real vs. Found Separatrices . . . . . . . . . . 46 3.2.2.4 Oset ...................... 46 3.2.2.5 Blocking the A-type Potassium Channel . . . 51 3.3 Discussion............................. 53 3.3.1 Model Separatrices . . . . . . . . . . . . . . . . . . . . 54 3.3.2 Separatrices in General . . . . . . . . . . . . . . . . . . 54 3.3.3 Sensitivity......................... 56 3.3.4 Pharmacology . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.5 Conclusion......................... 58 3.3.6 Outlook .......................... 59 Bibliography 60 A Supplements 64 A.1 Abstract.............................. 64 A.2 Zusammenfassung......................... 66 A.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 68 List of Figures 1.1 A spike in the 2D state space . . . . . . . . . . . . . . . . . . 9 1.2 The SIP and separatrix concepts . . . . . . . . . . . . . . . . 11 2.1 SIP Finder: rst snapshot . . . . . . . . . . . . . . . . . . . . 15 2.2 SIP Finder: second snapshot . . . . . . . . . . . . . . . . . . . 16 2.3 SIP Finder: nal snapshot . . . . . . . . . . . . . . . . . . . . 17 2.4 Found SIPs are plausible . . . . . . . . . . . . . . . . . . . . . 19 2.5 Dierences between rSIPs and fSIPs in models . . . . . . . . . 20 2.6 Spike dynamics in neurons and models . . . . . . . . . . . . . 21 2.7 The late spike phenomenon . . . . . . . . . . . . . . . . . . . 22 3.1 Stimulusscheme.......................... 32 3.2 Behavior of the modied leaky integrate-and-re model . . . . 34 3.3 The 4 types of separatrices . . . . . . . . . . . . . . . . . . . . 39 3.4 All measured separatrices . . . . . . . . . . . . . . . . . . . . 40 3.5 Neuron separatrices are oset independent . . . . . . . . . . . 42 3.6 Separatrix temporal stability . . . . . . . . . . . . . . . . . . . 43 3.7 The separatrix is a sensible indicator . . . . . . . . . . . . . . 45 3.8 Algorithm can reproduce real separatrices . . . . . . . . . . . 46 3.9 Oset dependence of model separatrices: LIF models . . . . . 47 3.10 Oset dependence of model separatrices: Wang-Buzsáki . . . . 48 3.11 Oset dependence of model separatrices: Awiszus . . . . . . . 49 3.12 Oset dependence of model separatrices: Hodgkin-Huxley . . . 50 3.13 Modeled A-type channel block . . . . . . . . . . . . . . . . . . 52 3 Nomenclature 2D Two-dimensional 3D Three-dimensional 4-AP 4-Aminopyridine ACSF Articial cereospinal uid AMPA Alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid AP Action Potential APV (2R)-amino-5-phosphonovaleric acid, a selective NMDA receptor antagonist CA cornu ammonis, a curved structure of densly packed neuronal somata within the hippocampal formation CGP-55845A [(2S)-3-[[(1S)-1-(3,4-dichlorophenyl)ethyl]amino]-2-hydroxypropyl] (phenylmethyl) phosphinic acid, a GABA B antagonist CNQX 6-cyano-7-nitroquinoxaline-2,3-dione, a competitive AMPA/Kainate receptor antagonist DG dentate gyrus, a V-formed structure of neuronal somata within the hippocampus fSIP found SIP: SIP found by using the forward/backward regression Method GABA Gamma-aminobutyric acid LIF leaky integrate-and-re NMDA N-Methyl-D-Aspartat PCI Peripheral Component Interconnect, a computer bus for attaching hardware devices to a computer 4 LIST OF FIGURES 5 rSIP real SIP: SIP determined by shortening the stimulus until no spike appeared. SEM Standard error of mean SIP spike initiation point, the point of no return within the cell's voltage trace. If the signal exceeds this point a spike is unavoidable. U Voltage, here the membrane voltage ˙ U rst time derivative of the voltage Chapter 1 Introduction Concerning the brain's information processing capabilities, neurons are the central elements of brain function. They communicate via action potentials (APs, also called spikes) sent to each other along the axons and transmitted through synapses. Despite the fact that more and more detailed knowledge erode many neurobiological dogmata it seems still generally correct to stick to the views that dendrite and soma are the input and processing regions for incoming signals, that the action potentials are generated from the membrane potential uctuations somewhere around the axon hillock, that spikes are the binary elements of the neural language, and that they are transmitted along the axon. Looking at the spikes as the key element of neural language, and focusing at the widely recorded intracellular membrane potential, this work will focus on the central question: what are the voltage conditions within a neural soma that elicit a spike? It is generally agreed that spikes are generated by a voltage threshold within the neuron, i.e. there is one xed threshold level the exceedance of which elicits a spike. However, it is also long known that the voltage threshold may vary in central neurons (e.g. see [44, 9, 5] and also [36]) as well as in distant elements such as the neuromuscular junction (e.g. see [38]). When looking at the spike generator from the viewpoint of dynamical systems mathematics, the variable U for the voltage would thus be its state variable, and the associated state space would then be one-dimensional. But the fact that the threshold varies within the neuron immediately indicates that the state space is too low-dimensional and demands the introduction of further state variables. 6 Chapter 2 A new Algorithm for Precise Detection of the Point of Spike Initiation 2.1 Materials and Methods The analysis procedures described here were performed on voltage data of action potentials, their time derivative as well as the appropriate stimulus signals. All signals were sampled with 35 kHz. They were recorded in vivo and simulated in silico according to the methods described in section 3.1. Please refer to that section for details on materials, recording, modeling or the like. 2.1.1 Finding the SIP As has been stated above (see 1.2.1), the SIP denes the transition from the passive pre-spike (i.e. sub-threshold) dynamic to the active (i.e. superthreshold) spike dynamic. The algorithm presented here will make use of the distinct features of the two types of dynamic in the U - ˙ U -plane. To nd the transition point, we will try to estimate it from within the pre-spike regime as well as from within the spike regime (see g. 2.1). Within the pre-spike dynamic, we take a 2.8 ms (100 data points at 35 kHz) long data window which is near the spike, but still clearly inside the passive regime. This is assured by dening a xed distance of 0.6 ms (20 data points at 35 kHz) from the spike's voltage maximum which is easy to detect. We calculate a linear regression from these data, thus projecting its trace beyond the SIP. This is justiable as the pre-spike dynamics is slow in relation to the 13 CHAPTER 2. PRECISE SIP DETECTION 14 Algorithm 2.1 Algorithm for the SIP nder START # finding the SIPs from a voltage trace Find spike maxima Convert voltage data to 2-D U-dU data Define data window relative to max(U) directly before, but clearly not inside spike Determine suitable extrapolation function through these data Define data window left of max(dU) spanning about 1/8 of the spike circle Determine linear regression function through these data For (each spike max) determine intersection between linear regression functions While (SIP not found) move spike window one data point backward determine actual intersection between linear regression functions If (actual intersection is right of last intersection) last intersection = actual intersection Else identify data point nearest to last intersection point SIP = identified nearest data point Endif Endwhile Append SIP to SIP-List Endfor Return SIP-List STOP short data window as well as in comparison to the spike dynamic, especially for the simple ramp stimuli used in this study. Within the spike regime, we start at max( ˙ U) and dene a similar window left from there with a length of 0.14 ms (4 data points at 35 kHz). We calculate the linear regression from this window as well and use it as a backward estimate for the spike dynamic. As the spike trace is nearly linear at spike start, this turns out to be a suitable assumption 1 . As an estimate for the SIP, we then calculate the intersection point of both regression lines which will lie far left at this rst step. In each subsequent iteration, the in-spike regression window is shifted left (i.e. backward in time) one data point at each step while the window size is kept constant. As a consequence, the in-spike regression line will become steeper and steeper, and the intersection point will shift right on the pre-spike regression line with each iteration. With the in-spike data window moving backward in time, it will eventually enter the pre-spike regime, thus resulting in a shallower regression line and thus in the intersection point shifting left again. The algorithm stops here, returning the rightmost intersection point as the best estimate for the SIP. This estimate is then mapped to the real data by choosing the data point with the smallest euclidean distance to the SIP estimate. 1 More sophisticated ts have also been tested, but they turned out to be too prone to slight variations in spike onset dynamics. CHAPTER 2. PRECISE SIP DETECTION 15 -50 -40 -30 -20 -10 0 50 100 150 200 -41.5-41-40.5-40-39.5-39-38.5-38 -4 -2 0 2 4 6 8 10 Figure 2.1: SIP Finder: rst snapshot. This gure illustrates an early stage of the SIP nder process (see gures 2.2 and 2.3 for subsequent snapshots). Within all 3 gures the left plot displays the spike onset in the state space, with the dense dots resembling the passive pre-spike dynamics followed by the start of the (active) spike trajectory. The right plot in each gure shows a magnied view of the transition point. The gray points are the data clearly within the pre-spike dynamic which are used to calculate the linear pre-spike regression (blue line, blue dots are the calculated supporting points). The magenta-colored points are clearly within the spike dynamic, their linear regression is the green line (green dots are the calculated supporting points). The red dot is the current intersection of both regression lines, the yellow dots resemble the previous intersection points. The red circle denotes the previous intersection point relative to the current one. U [mV] is on horizontal axis, ˙ U [mV/ms] at vertical axis. CHAPTER 2. PRECISE SIP DETECTION 16 -50 -40 -30 -20 -10 0 50 100 150 200 -41.5-41-40.5-40-39.5-39-38.5-38 -4 -2 0 2 4 6 8 10 Figure 2.2: SIP Finder: second snapshot. This gure shows the next step after gure 2.1. The in-spike regression window has been shifted backward in time by one data point. The resulting green regression line intersects with the blue pre-spike line more right than before (see red dot). The previous intersection point (red dot in left gure2.1) is now yellow, marked with a red circle. With each step the intersection point shifts more and more right on the blue pre-spike regression line. See gure 2.1 for gure legend. CHAPTER 2. PRECISE SIP DETECTION 17 -50 -40 -30 -20 -10 0 50 100 150 200 -41.5-41-40.5-40-39.5-39-38.5-38 -4 -2 0 2 4 6 8 10 Figure 2.3: SIP Finder: nal snapshot. This gure illustrates the nal stage of the SIP nder sequence. The right gure is the last step in the process: now the left boundary of the in-spike regression window has reached the pre spike dynamic, resulting in a shallower green regression line. This causes the regression intersection to be more left on the blue regression line (compare red dot with red circle denoting the last intersection point. This causes the process to stop, and the rightmost intersection point is taken to be the rst estimate for the SIP. See gure 2.1 for gure legend. CHAPTER 2. PRECISE SIP DETECTION 18 2.1.2 Checking the SIP nder In order to verify the precise function of the SIP nder we need an alternative way to determine the transition point from pre-spike to spike dynamic. As this point is dened as the point-of-no-return we can nd it by making the stimulus ramp shorter and shorter until no spike is elicited any more. This would thus mean to switch o the stimulus exactly at the SIP. Despite all eorts for a clean adjustment of electrode and recording parameters artifacts in the recorded voltage trace (due to the sharp edge at stimulus switch-o) could not be avoided. As the delicate analysis of the SIPs would have been thoroughly aected by these artifacts this procedure was carried out solely on models. In this approach the SIP is searched for using the binary search algorithm. The iterative procedure starts with a ramp stimulus of an initial length L0 . In the next steps the ramp is either elongated (if the last iteration did not elicit a spike) to Ln+1 =Ln+Ln 2 or shortened (if the last iteration did elicit a spike) to Ln+1 =Ln−Ln 2 for iteration n with n= 0 at the initial step. This is repeated until a temporal resolution of ∆t50.001 ms is reached. These real SIPs (rSIPs) can then be compared to the nder's SIPs (fSIPs) by calculating the temporal dierence between both points. Using the separatrix concept presented in chapter 3 we can also analyze the temporal distance between complete separatrices. 2.2 Results 2.2.1 SIP Finder For all natural spikes analyzed we can precisely and robustly determine the phenomenological SIP with the algorithm described here. Precisely means that the fSIPs - the only ones available for natural spikes - are both plausible in U and ˙ U as well as very close to the pre-spike subthreshold dynamic (see g. 2.4). Robustly means that the SIP nder is able to nd the SIP in all cases of healthy spikes. With the stimulus shortening procedure using the simulated models we have an alternative tool at hand (see 2.1.2) to determine the real SIPs. Comparing these with the found SIPs, we encounter large dierences between rSIPs and fSIPs (up to 1.3 ms) with most Hodgkin-Huxley type models used here. This means that reaching the rSIP does induce the irreversible dynamic that will lead to a spike, but obviously in the beginning this dynamic is much too weak to result in any detectable eects in the membrane voltage. CHAPTER 2. PRECISE SIP DETECTION 19 -4 -3 -2 -1 0 -60 -40 -20 0 20 -60 -40 -20 0 20 0 100 200 300 Figure 2.4: Found SIPs are plausible. SIP location is plausible in U (left plot), ˙ U as well as in the U / ˙ U state space (right plot). Colors code from red via orange, yellow, green to blue for steepness of stimulation ramp: red means shallow ramps, blue are steep ramps. Black dots indicate position of detected SIPs. Left plot shows voltage [mV] against time [ms], spikes are aligned with their voltage maximum at t=0. Same group of spikes is shown in right plot in state space with U [mV] on the horizontal axis and ˙ U [mV/ms] on the vertical axis. CHAPTER 2. PRECISE SIP DETECTION 20 -4 -3 -2 -1 0 -60 -40 -20 0 20 40 fSIPs rSIPs -60 -40 -20 0 20 40 -200 0 200 400 600 fSIPs rSIPs Figure 2.5: Dierences of rSIPs to fSIPs in models. This Plot shows the Wang/Buzsáki variation of the Hodgkin/Huxley type model, answering to ramp stimulation. Colors code from red via orange, yellow, green to blue for steepness of stimulation ramp: red means shallow ramps, blue are steep ramps. Black dots indicate position of detected SIPs: while the right-hand cloud of SIPs indicate the fSIPs in both plots, the lefthand cloud in the right plot as well as correspondingly the left-hand stripes of dots in the left plot show the rSIPs. Left plot shows voltage [mV] over time [ms], spikes are aligned with their voltage maximum at t=0. Same group of spikes is shown in right plot in state space. U [mV] is on horizontal axis, ˙ U [mV/ms] at vertical axis. CHAPTER 2. PRECISE SIP DETECTION 21 -60 -40 -20 0 20 40 -50 0 50 100 150 200 250 300 -60 -40 -20 0 20 -400 -200 0 200 400 600 800 -60 -40 -20 0 20 0 100 200 300 Figure 2.6: Spike dynamics in neurons and models. Left: Trajectories of the original Hodgkin-Huxley model. Spike onset is very shallow. Middle: Trajectories of the modied Awiszus model. Note that spike onset dynamic is much faster here. Right: Trajectories of a real neuron. Spikes are still faster at spike onset, the trajectories emerge nearly vertical from the pre-spike dynamic. See gure 2.5 for the eects of kinkiness on real and found SIPs. Luckily, we see an interesting eect among the models. Contrary to the other Hodgkin-Huxley models tested, the modied Awiszus model does detect the fSIPs much close to the rSIPs (see g. 3.10). This model phenomenologically diers from the other ones in so far as its spike dynamic shows a much more abrupt onset. Close inspection of the trajectories of models and neurons reveal signi- cant dierences concerning the speed of the dynamic at spike onset (see g. 2.6). We can see that the faster the spike dynamic at spike onset, the closer the fSIPs are to the rSIPs. As the modied Awiszus model has a much faster dynamic than the original Hodgkin-Huxley model, it allows the detection of the fSIPs much closer to the rSIPs. Comparing the trajectories of a real neuron to those of the models, we see that it is still much faster at spike onset than any of the Hodgkin-Huxley type models. This justies the interpretation that the fSIPs found on the neuron's trajectories are similar to the rSIPs we cannot precisely detect. 2.2.2 Late Spike Phenomenon During the shortening of the stimulus ramps we encountered an interesting phenomenon in all Hodgkin-Huxley type models. When the ramp was shortened to determine the precise rSIP, it occurred for certain slope/duration combinations that a spike occurred far (up to 60 ms) after the stimulus had been switched o. CHAPTER 2. PRECISE SIP DETECTION 22 0 200 400 600 800 1000 -20 0 20 40 60 80 100 0 200 400 600 800 1000 -20 0 20 40 60 80 100 0 500 1000 1500 2000 2500 3000 3500 -80 -60 -40 -20 0 20 0 500 1000 1500 2000 2500 3000 3500 -80 -60 -40 -20 0 20 Figure 2.7: Late spike phenomenon. Top left: a ramp of length 5.23438 ms and slope 758 mV/ms in the classic Hodgkin-Huxley model induces a spike which occurs far after the ramp has been switched o. This illustrates the concept of a point of no return as a denition of the SIP. Top right: a slightly shorter Ramp (5.2334 ms) does not produce a spike but only a bump instead. The late spike phenomenon depends on the parameter set of the respective model. The lower two gures show a much longer time delay for the Wang/Buzsáki variation of the Hodgkin/Huxley type model. Ramp parameters are: Length 4.29688 ms and slope 758 mV/ms (lower left), length 4.2959 ms with the same slope (lower right). All gures: red curve shows stimulus without scale for timing information only. Grey rectangle maps beginning and end of stimulus to the voltage trace. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 29 the upper end through which carbogenated ACSF was supplied, and a sink on the opposite side. The ACSF feeding pipes lead through the water below, thus warming up the ACSF. Each experimenting chamber was equipped with two AgCl pellets as grounding electrodes. The interface chamber was xated by permanent magnets on the heavy steel top of a active pneumatic experimenting table (Science Products GmbH, Hofheim, Germany) which isolated the setup from vibrations. The slice was illuminated by a cold light source (Olympus, Tokyo, Japan) via ber optics. A stereo microscope (Leica Microsystems GmbH, Wetzlar, Germany) provided optical control of slice quality and coarse electrode position. Before starting the experiment, the oor of the slice chamber was covered with one layer of lens cleaning paper (62647-B, Kodak, Rochester, New York, USA). For drainage, the inner chamber walls were aligned with 2 mm wide stripes of nylon stocking fabric which also led into the chamber's sink. Little sheets of Kodak lens cleaning paper of an appropriate layout ensured that the reference electrode pellets were in good electrical contact to the ACSF. The chamber was constantly own through by carbogenated and warmed ACSF, supplied by a peristaltic pump (Minipuls 3, Gilson Inc., Middleton, UK) via the two nozzles at a rate of about 1.6 ml/min. For experiments, slices were transferred into the experimenting chamber on their little sheets of lens cleaning paper using a pair of tweezers and placed near to the ACSF supplying nozzle. For pharmacological experiments, the ACSF hose was removed from the standard ACSF reservoir and placed inside a second ACSF reservoir with added drugs. It took about 3 minutes for the new solution to reach the chamber's nozzle. Once a week, all components of the setup that come into contact with ACSF (hoses, storage chamber, experimenting chamber) were cleaned using a 0.3 M solution of H 2 O 2 . 3.1.2.5 Cell Search and Stimulus Parametrization The electrode was positioned over the CA1 region where all cells were recorded. It was then lowered until electrical contact with the tissue was established. The bridge balance was adjusted at the amplier to correct for the electrode's resistance. The capacity compensation was adjusted to correct for the electrode's capacity. During cell search, the electrode was lowered in steps of about 2 nm, using a mechanical 3D positioning device (Leica Microsystems GmbH, Wetzlar, Germany). To clean the electrode tip as well as to make cell penetration easier, the so-called buzz 1 was used frequently. This cell 1 Buzz means a short increase of the capacity compensation, yielding to an oscillation of the compensation circuit. The processes leading to both an easier cell penetration as well as cleaning the electrode tip is not known, but it is a successful standard procedure CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 30 search procedure was done without microscopic control, i.e. blind. Bridge balance and capacity compensation were permanently checked and adjusted if necessary. While in the extracellular space, a repetitive hyperpolarizing pulse (amplitude 0.1 nA, duration 100 ms) was used with bridge balance slightly out of balance. Proximity of a cell was suspected when the apparent electrode resistance showed a sudden increase or when extracellular AP appeared. After penetration of a cell it was hyperpolarized in order to help the cell to recover from penetration. Under permanent control of the cellular health state the hyperpolarizing current was slowly reduced to 0. Cells were then given about 10-15 minutes time for accommodation and regeneration before starting any stimulation. After this pause, they were roughly checked for ring threshold, resistance, time constant and spike amplitude. If these parameters seemed healthy (see 3.1.4.1 for healthy parameters), the cells were used for experiments. First, the amplitudes of the depolarizing and hyperpolarizing oset currents were identied. The depolarizing oset amplitude was chosen to be slightly sub-threshold. The hyperpolarizing oset was chosen to be roughly 5-10 mV below the resting potential. Now the slopes of the stimulus currents were dened from the resting potential. As the precise form of the membrane potential during a ramp could not be controlled, the membrane potential before the stimulus and the potential at spike initiation were used as as xed points and a linear estimate was made. Stimulus slopes were now chosen such that the resulting estimated linear membrane potential slopes were roughly 0.1, 0.2, 0.4, 1.0, 2.0 mV/ms. This procedure was repeated for both osets potentials in order to investigate the inuence of the oset on the threshold. 3.1.2.6 Stimulation Stimulus generation and data recording was controlled by an individually developed LabView program. It provided a graphical user interface for fast, easy and exible access to all necessary parameters and controls. It also provided automatic mechanisms to assure that all relevant recording and stimulation parameters were saved in the data le header. The computer-generated stimulus signals were digitized at 35 kHz in order to ensure high temporal precision even with fast signals. They were output via a data interface card (Type PCI MIO 16 E 4, National Instruments Corp., Austin, Texas, USA) and then amplied to comply with the needs of the for sharp electrode recordings. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 31 stimulus channel of the amplier (Model IR-183, Neuro Data Instruments Corp., New York, USA), fed into it and applied to the cell through the recording electrode. Additionally, this stimulus output of the Neuro Data amplier was recorded by the computer again (via the same card) and saved along with the cellular signal (one individual le per stimulus). Thus, the real stimulus is accessible at any part of the intracellular signal. Both the stimulus and the cellular signal were also displayed on an oscilloscope for online control. The recorded intracellular membrane potential was fed into the Neuro Data amplier via its high-resistance (about 1013 Ω ) head stage pre-amplier, low-pass ltered at 3 kHz, digitized at 35 kHz via the PCI card and stored on the computer's hard disk (Pentium III 700 MHz Processor, Windows NT). This high sampling rate was chosen in order to be able to precisely locate the start of the spike. As the spike start contains relatively low frequencies as compared to the lter's cut-o frequency, this does not interfere with the low-pass ltering. 3.1.2.7 In Vitro Experiments After identication of all stimulus parameters for the individual cell the experiments could be started. At rst the cell was stimulated by a 500 ms long positive square pulse, adjusted to an amplitude that initiated a sequence of spikes. In data analysis later on this served as an indicator for the cell type. Second, a sequence of 9 dierent 200 ms long square pulses with dierent sub-threshold positive and negative amplitudes was given, thus allowing to determine voltage/current relationship, the cell resistance and the time constant. For each experimental run there were 5 dierent slopes to be tested at 3 dierent osets (positive oset, resting potential and negative oset), i.e. 15 parameter combinations. Each parameter combination was given 10 times to enhance reliability, and the sequence of parameter sets was pseudo-randomly selected. There was a pause of at least 2 s between each parameter set to avoid any aftereects of the previous stimulation (see g. 3.1). In case of stability analysis, this procedure was repeated after a pause in order to investigate temporal stability of the results. In case of pharmacological experiments, Drugs were added to the ACSF and perfusion was started at least 20 min before the actual measurement since diusion into the slice is rather slow (see [35]). After that the procedure was repeated with the same stimulus parameters. After a pharmacological experiment, the slice was discarded and the chamber thoroughly washed with standard ACSF before using a new slice. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 32 10 ms 20 mV 1 nA Figure 3.1: Scheme of a typical stimulation. Lower trace shows the stimulus, starting (in this case) with an negative oset (left) of -100 pA. Upper trace shows the cell's answer. One stimulus run had the constant length of 1 second, which consisted of a 300 ms delay after the onset of an eventual oset, the ramp itself, and a trailing pause after which the oset was switched o (right side of gure). Between two stimuli there was a 2 second pause. Specifying an optional boundary allowed to start recording before and after the stimulus, i. e. before and after the oset. Typical stimulus slopes used were between 1.5 and 250 nA/s. Stimulus amplitude was 744 pA in this case. For pharmacological experiments, the idea was to test the impact of the A-type potassium current on the ring threshold. In order to test this hypothesis, these channels were blocked using 50 µ M 4-Aminopyridine (4-AP). Application of 4-AP with 50 µ M aects (among others) K currents mediated by Kv1 and Kv3 channel members and can induce seizure like events. To prevent generation of epileptiform discharges synaptic transmission was blocked using a cocktail of glutamate and GABA receptor antagonists (30 µ M CNQX (6-cyano-7-nitroquinoxaline-2,3-dione, a competitive AMPA/Kainate receptor antagonist), 60 µ M APV ((2R)-amino-5-phosphonovaleric acid, a selective NMDA receptor antagonist), 5 µ M Bicucullin (a competitive GABA A antagonist) and 1 µ M CGP-55845A ([(2S)-3-[[(1S)-1-(3,4-dichlorophenyl)ethyl]amino]-2-hydroxypropyl](phenylmethyl) phosphinic acid, a GABA B antagonist)). 3.1.3 Models 3.1.3.1 Modied Leaky Integrate-And-Fire Model The leaky integrate-and-re (LIF) model was introduced by Lapicque in 1907 [29]. Contrary to the models of the Hodgkin-Huxley type, the LIF model is a simple one. It incorporates a constant capacity, a constant resistance and resembles a low-pass lter: CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 33 C˙ U+1 R(U−U0) = I This term computes the passive uctuations of the membrane potential. In order to use it as a neuron model a threshold criterion is introduced: U=U0 if U≥Uθ Typically, it is equipped with a xed voltage threshold, and its membrane potential is simply reset to a given value U0 when the threshold is reached, i.e. no spike is produced. This is a problem for the questions to be investigated here as they explicitly need to look at the voltage trace exactly at the transition from the sub-threshold behavior to the spike behavior. Especially the SIP nding algorithm needs the spike upstroke to determine the SIP. In order to cope for this, an articial analytical spike is attached to the model's voltage trace exactly at U(t) = Uθ (see left plot in g. 3.2). Of course explicit care is taken to assure that there is a smooth transition from the model trace to the spike trace as this will be the point to be analyzed later. To do so, we determine the slope ( dU dt ) of the voltage trace at U(t) = Uθ and nd the attachment point with exactly the same slope within the onset of the articial spike. The articial spike is an analytical function that has been precisely tted to the onset and upstroke of a real spike according to the procedure described in [18]. It mimics the spike onset and upstroke in great detail (see right plot in g. 3.2). In order to adopt it to the spike features ( U and ˙ U amplitudes) of the data at hand it was compressed in the time domain by factor 2. This modication was used solely to make results comparable to the cellular data. Within this simple model we are able to predene arbitrary separatrices. We will thus be able to test the analytical algorithms used further on to check whether they are able to reproduce these built-in separatrices. Three dierent threshold criteria were used for the leaky integrate-and- re models: 1. Type A (exponential): UΘ=(1000 ˙ U≤0 exp 3−0.2˙ U otherwise 2. Type B (root): UΘ=   1000 ˙ U≤0 r20 ˙ U+ 1 otherwise 3. Type C (quasi linear): UΘ=(1000 ˙ U≤0 5+0.5˙ U otherwise CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 34 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 0 10 20 30 40 50 60 Figure 3.2: Behavior of the modied leaky integrate-and-re model. Left: when stimulated, the membrane potential of the model rises to the threshold value Uθ . While the original model would now simply reset the voltage to U0 , an analytically dened spike is attached to the voltage trace. It is taken care to assure that dU dt is exactly identical at the transition from model trace to spike trace. As we are only interested in the spike initiation we do not need to take care of the post-spike behavior of the model. Right: The analytical spike that is attached to the model. The slope of the point of attachment on the spike is chosen to be exactly the same as on the model's voltage trace at U(t) = Uθ . Both gures show voltage [mV] against time [ms]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 35 For easier comparison with cells and the other models the calculated voltage data of the leaky integrate-and-re models were shifted to the mean membrane potential of all measured cells before analysis. 3.1.3.2 Hodgkin-Huxley-Model The original Hodgkin-Huxley model ([24], used in the version from [28]) resulted from measurements that Hodgkin and Huxley had carried out on the giant axon of the squid. This and the fact that the measurements were done at low Ca + and at a temperature of 10 ° C indicate that there is a big dierence in model behavior compared to a mammalian neuron at body temperature. Nevertheless, the Hodgkin-Huxley model is modeled because of its principle importance as a reference. The central equation of the Hodgkin-Huxley model is the current balance equation CmdV dt =INa+IK+Ileak +Iinj, that is it basically depends on the sum of all currents owing in and out of the membrane, according to Kirchho's law. There is the sodium current INa and the potassium current IK , both resembling the currents that ow through the corresponding ion channels, the leak current Ileak , indicating that the membrane is not hermetically sealed, and of course all currents injected into the cell via a stimulation electrode, Iinj . For better comparison with cells and the other models the calculated voltage data of the Hodgkin-Huxley model was shifted to the mean membrane potential of all measured cells before analysis. 3.1.3.3 Wang-Buzsáki-Model In 1996, Wang and Buzsáki [45] introduced a Hodgkin-Huxley type model of hippocampal interneurons. Their motivation was to investigate gamma oscillation in a hippocampal network model, so their current balance equation is CmdV dt =I Na +I K +I syn +I leak +I inj with the synaptic input current I syn . The modications of the original Hodgkin-Huxley model were made in order to phenomenologically adjust the model behavior to the behavior of real interneurons in respect to an afterhyperpolarization and high ring rates. There is no need, however, to consider any synaptic input for this work, so the model falls back to the current balance equation known from the standard Hodgkin-Huxley model. The parametrization is, of course, dierent. 3.1.3.4 Awiszus-Model In 1992, Awiszus [1] presented a modied Hodgkin-Huxley model. His intention was to adopt it to the behavior of a mammalian neuron at body CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 36 temperature. Basis for this modication was voltage-clamp data from small rat neurons in the supra-optic nucleus area [15, 14]. In this modication, an A-type potassium channel is added to the standard Hodgkin-Huxley model, so that the basic equation is CmdV dt =I Na +I K +I A +I leak +I inj . This family of models will allow to test the impact of the A-type potassium current. 3.1.3.5 Reduced Awiszus-Model In his paper [1] Awiszus reduces the model from 6 to 5 dimensions by utilizing the close relationship of the variables a and h . In this 5-dimensional model, the corresponding equations are combined into one, thus reducing the dimensionality. All other equations are identical with the full model. 3.1.3.6 Modied Awiszus-Models With the full Awiszus model modeling the A-type potassium channel, we have the ability to investigate the inuence of this type of channel on the separatrix. We have thus modied the model by blocking the A-type potassium current. This is done by setting the maximum conductance for the A-type potassium channel to G A = 0 mS/cm 2 . As this modication resulted in a permanently spiking behavior at resting potential, we adjusted the leak current parameters to G leak = 0.1768 mS/cm 2 and E leak =−76.95 mV such that the model did not spike at rest. This model will be referred to as NoA. In order to provide a usable comparison for this modication we modied the full model to the same values for G leak and E leak . 3.1.3.7 In Silico Experiments The models were stimulated in the same way as the biological cells. Calculations were carried out on x86 computer architecture running Linux operating systems (RedHat Version 7.1 and 7.3, Suse Version 9.0, 9.1, 9.2, Ubuntu Version 8.9 and 9.1), using the Mathematica software package (Wolfram Research, Inc.) in versions 4 and 5. Models were numerically solved using the built-in numerical solver for dierential equations with a maximal step size of 0.1 s. Preliminary tests were conducted to check the inuence of external noise on the separatrix. For these we recorded intracellular noise with the experimental setup and shifted its mean to 0. This noise was then used to noise the models by simply adding it onto the smooth model results. Further comparison between pure and noisy models revealed identical SIPs (and thus identical separatrices), so all further analysis was done with pure (i.e. noise-free) models. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 37 3.1.4 Data Analysis 3.1.4.1 Cell Quality Criteria Cells were only used for further analysis if they met the following quality criteria:  spike amplitude more than 60 mV  action potential overshooting  membrane resistance more than 25 M Ω  membrane time constant more than 2.5 ms 3.1.4.2 Data Postprocessing The postprocessing procedure consisted of several self-developed Perl and Mathematica tools for automatized data preparation, arrangement and feature extraction. The aim of this process was to prepare and arrange the crucial data for easy, standardized and widely automatized access by the analysis programs. The following steps were applied: 1. unzip data le 2. read parameter header 3. nd maximum of rst spike during stimulus ramp 4. write relevant information into log le 5. re-zip data le Using a self-developed Mathematica tool set, this extracted information was used to extract the relevant data (spikes as well as stimulus) from the raw data les and save them in a binary Mathematica format for easy and standardized access for all further analysis. 3.1.4.3 Data Analysis All data have been processed in the same way, independent of their origin (cells or models). First of all, only the rst spike on each ramp was used for further analysis in order to eliminate any spike aftereects at the SIP. This spike was extracted from the original data le together with the corresponding stimulus trace. From this original voltage data the voltage derivative was calculated using the mean of three subsequent data points: CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 38 ˙ Ui=f 1000 Ui+1−Ui−1 2 with ˙ Ui , the derivative of the i th data value, Ui , the voltage value of the i th data value, f , the sampling frequency, and i , the index. With the voltage given in mV this calculation yields the derivative in units of mV/ms. It is a simple algorithm for calculating the derivative for evenly spaced data, and it only makes a minor error within the 3-value data window. We also have tested more complex derivative lters like e.g. the Savitzky-Golay lter class, but there was no signicant dierence within the results. Having U and ˙ U , we can now use the algorithm described in 2 to precisely determine the SIPs of the spikes. All 10 SIPs generated with the same parameter combination were grouped together, and the center of gravity (2-dimensional mean) of each of these clouds was calculated together with the 2-dimensional standard deviation and standard error of mean (SEM). Models of course were calculated only once per parameter combination as they are completely deterministic. We tted a function of the form a0+ a1x+a2log(x) with an∈R to each state variable separately by minimizing the 2-dimensional distance of the function to the means. These tted 2dimensional function now is the separatrix of the cell. 3.2 Results 3.2.1 Cells 3.2.1.1 Separatrices We have been able to record from 22 cells that met the quality criteria (see 3.1.4.1). Among these cells were 10 regular spiking, 5 oscillatory, and 6 fast spiking. One cell could not be clearly allocated to a special cell type by the recorded data. The cells showed very dierent types of separatrices. They can be grouped into 4 groups:  vertical (n=6): The vertical separatrix is the type of threshold one would expect from a cell with a pure voltage threshold. It varies in the ˙ U domain, but all spikes are elicited at the same voltage U .  horizontal (n=4): This is the opposite of the vertical separatrix. All spikes start at the same ˙ U value, so eectively we have a ˙ U threshold. As a consequence, the SIPs span up to 20 mV, thus again strongly questioning the 1-dimensional threshold concept. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 45 -28 -26 -24 -22 -20 -18 -16 2.5 5 7.5 10 12.5 15 17.5 20 -52.5-50-47.5-45-42.5-40-37.5-35 2.5 5 7.5 10 12.5 15 17.5 20 -60 -55 -50 -45 -40 -35 -30 2.5 5 7.5 10 12.5 15 17.5 20 Figure 3.7: The separatrix is a sensible indicator of the cell's state. Left plot: The separatrix seems to be a very sensible indicator of the cell's physiological state. This fast spiking neuron initially showed the red separatrix without any pharmacology. Second, the green separatrix was measured 30 min after application of the synaptic block cocktail without 4-AP. This already shifted the separatrix 3 mV to the left. Third, the additional application of 4-AP did not change it from its shifted state. Additionally, the following cell parameters changed between runs: membrane resistance +5 / +1 M Ω , spike amplitude -4 / ± 0 mV. Middle plot: A regular spiking neuron. Red: The initial separatrix. Green: After 20 min under block+4-AP. Blue: After 50 min under block+4-AP. Additionally, the following cell parameters changed between runs: membrane time constant -1 / -2 ms, membrane resistance -4 / -12 M Ω , spike amplitude +4 / ± 0 mV. C: Fast spiking neuron with its initial red separatrix, the green after 35 min under block+4-AP, the turquoise after 70 min under block+4-AP, the blue after 35 min wash. Additionally, the following cell parameters changed between runs: membrane time constant -2 / ± 0 / -2 ms, membrane resistance +1 / +1 / +4 M Ω , spike amplitude ± 0 / +1 / -11 mV. Crosshairs show the SEM of the means, projected onto the separatrices, the means of the SIPs are omitted for better visibility. Crosshairs of left and right plots have been omitted for the same reason. Colors of crosshairs code for the slope of the stimulus from shallow (red) to steep (blue). Colors of separatrices in all 3 plots are solely for better distinguishability and do not code for any parameter. Axes show ˙ U [mV/ms] against U [mV]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 46 -60-57.5-55-52.5-50-47.5-45 2.5 5 7.5 10 12.5 15 17.5 20 -60-57.5-55-52.5-50-47.5-45 2.5 5 7.5 10 12.5 15 17.5 20 -60-57.5-55-52.5-50-47.5-45 2.5 5 7.5 10 12.5 15 17.5 20 Figure 3.8: Algorithm can reproduce real separatrices. Real vs. reconstructed separatrices of the 3 dierent predened separatrix types in the LIF models (see 3.1.3.1). Left: type A, middle: type B, right: type C. The small deviations of the reconstructed separatrices are due to the limited temporal resolution because of the sampled model signal as well as to a systematic eect of the SIP nder (see 2). The leftmost separatrix is the real separatrix, while the found one is slightly more right. However, they are near enough to show that the algorithm presented here is able to phenomenologically reproduce a built-in separatrix. detection of the SIPs and thus the separatrices. However, as the state space is spanned by U and ˙ U , any distortion of the voltage trace before the SIP will simply shift the SIP along the separatrix without altering its form. 3.2.2.3 Real vs. Found Separatrices Using the models, we are able to determine the real SIPs (and thus the real separatrices) via a second method (see chapter 2). Depending on the abruptness of the spike onset we see a signicant divergence between real and found separatrices. Please refer to 2.1.1 for the results concerning real vs. found SIPs. 3.2.2.4 Oset The separatrices of the modied Leaky Integrate-and-Fire models are - by denition - independent of the oset. Of course dierent osets lead to dierent SIPs, especially because identical stimulus slopes lead to dierent voltage slopes at the SIPs. However, as the predened threshold is independent of any other parameters but U and ˙ U , these eects merely shift the SIPs along the separatrix (see the SEM ellipses elongated along the separatrix in g. 3.10). CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 47 Figure 3.9: Oset dependence in LIF models. The separatrices of the leaky integrate-and-re models are per denitionem independent of the oset. This Figure shows the real separatrix of the type A model (see 3.1.3.1), with all osets grouped together. Shifts occur at the steeper parts of the separatrix, but the orientation of the standard deviation ellipses show that they follow the separatrix itself, thus retaining its form. Dots denote the SIPs for the dierent stimulus ramp slopes, with the color shifting from red (shallow ramps) to blue (steep ramps), ellipsoids show the SEM. Axes are ˙ U [mV/ms] against U [mV]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 48 Figure 3.10: Oset dependence of the Wang-Buzsáki model separatrices. The Wang-Buzsáki modication of the Hodgkin-Huxley model as an example for the oset dependence of the separatrix. The three distinct lines in the bottom left corner show the real separatrices, found via the ramp shortening paradigm (see 2.1.2). The upper right corner shows the three corresponding found separatrices, determined by the SIP detection algorithm presented in 2. The increased variability of the SIPs at the found separatrices is due to the much faster dynamic at this point. Dots denote the SIPs for the dierent stimulus ramp slopes, with the color shifting from red (shallow ramps) to blue (steep ramps). Axes are ˙ U [mV/ms] against U [mV]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 49 Figure 3.11: Oset dependence of modied Awiszus model separatrices. The modied Awiszus model in the same layout. Note that the three detected separatrices are close to the real separatrices. Also note that the orientation and overall form of the found separatrices are identical to the real separatrices, although they are shifted in state space. Dots denote the SIPs for the dierent stimulus ramp slopes, with the color shifting from red (shallow ramps) to blue (steep ramps). Axes are ˙ U [mV/ms] against U [mV]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 50 Figure 3.12: Oset dependence of Hodgkin-Huxley model separatrices. The original Hodgkin-Huxley model shows a similar, but slightly dierent behavior. Instead of the three oset separatrices being shifted along the trajectories they are shifted perpendicular. However, we again nd the found separatrices to be shifted along the trajectories, thereby reproducing the real ones. Dots denote the SIPs for the dierent stimulus ramp slopes, with the color shifting from red (shallow ramps) to blue (steep ramps). Axes are ˙ U [mV/ms] against U [mV]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 51 The Hodgkin-Huxley models are clearly sensitive to osets, independent of their avor. Within all of these models we see the three real separatrices for positive, zero and negative osets clearly separated. For the original Hodgkin-Huxley model the dierent osets shift the separatrices perpendicular to the course of the trajectories (see g. 3.10 C), with the negative osets inducing the upper and positive ones inducing the lower separatrix. This again also means that in this case the oset shifts the course of the trajectories. For all other Hodgkin-Huxley avors tested here, the separatrices are shifted along the trajectories. Contrary to the original model this indicates that the oset does not change the course of the trajectories. In these models the rightmost separatrix is elicited from a hyperpolarized model, the middle one from a model at resting potential, the leftmost from a depolarized model. All models however show that the overall form of the real separatrix is retained in any case, independent of the oset. The separatrices found by the SIP detector (see 2) typically lie far more right (i.e. shifted forward in time along the trajectories) than the real ones, meaning that they are detected signicantly after their real occurrence. Only for the modied Awiszus model we nd the detected separatrix within the range of the real separatrices. The temporal dierence between real and found separatrices is directly linked to the abruptness of the spike emerging from the pre-spike dynamics: the smoother (i.e. less kinky) the spikes emerge (see g. 2.5 for a comparison), the more distant the found separatrices are from the real ones, and vice versa. It is worth noting again that the phenomenological separatrices found by the algorithms and procedures presented in this work are able to reproduce the key features of the separatrices in any case, be it angle, extension and curvature. As the amount of divergence of real to found separatrices is clearly linked to the sharpness of the trajectory's kink at spike onset, this again nourishes the view that our algorithm can be able to precisely identify the separatrices of real neurons as well. 3.2.2.5 Blocking the A-type Potassium Channel With the A-type current being explicitly available in the Awiszus model, we can test the model's response to completely blocking the A-type current. To do so we have to compare the modied Awiszus model (NoA) with the modied Awiszus model with implemented A-type current as these two dier only in their A-type channel conductance. While the found and real separatrices are rather close for the full model they are again clearly separated for the NoA model. As another support of our nding that the proximity of real and found separatrices is linked to the CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 52 -66 -64 -62 -60 -58 0 5 10 15 20 real w/o a real with a found w/o a found with a Figure 3.13: Modeled A-type channel block. The red and green separatrices are the found and real ones of the control model with working A-type channel, the turquoise and blue separatrices belong to the model with blocked A-type channel. The comparison of the real separatrices of the two models (green vs. blue) shows a signicant left-anddown shift (-5 mV, -4 mV/ms) together with a straightening. The found separatrices are in close proximity and show just a small left-and-up shift (-1.5 mV, +1.5 mV/ms). Note that the overall angle of tilt is preserved for all 4 separatrices. The gray areas indicate the approximate corridor of the trajectories bundle. The waved area interconnects the two separatrices of the model without the A-type current, and the checkered one interconnects the two separatrices of the model with the A-type current. The checkered corridor is shifted a little to the lower right side, thus indicating that the model with the A-type current shows a slightly slower dynamic. Dots show the means of the SIPs elicited by the same stimulus slope. Colors of dots code for the slope of the stimulus from shallow (red) to steep (blue). Colors of separatrices are solely for better distinguishability and do not code for any parameter. Axes are ˙ U [mV/ms] against U [mV]. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 53 abruptness of the spike onset, the full model turns in deed out to have a faster spike onset than the NoA version. Looking at the found separatrices, we nd a slight right-shift as well as a slight down-shift for the NoA Version as compared to the full model. This result is similar to the results found for the neurons (see section 3.2.1.4). However, the neurons showed this eect already when the synaptic block cocktail was added, there was no systematic 4-AP-related eect visible. A more reliable result for the models is of course the real separatrix. Here we see a clear shift of about 5 mV to the right and about 3 mV/ms upward. The increased variability of the rSIPs (see green separatrix in g. 3.13) indicates that the spike dynamic becomes irreversible later, closer to the fSIP separatrix. The real separatrix becomes curved, but maintains its overall orientation. Looking at the corridors of the trajectories bundle, it seems that the separatrix is not detected earlier, but the models spike dynamic becomes later irreversible. However, due to obscure pharmacological results with neurons we are unable to link these ndings to biology. 3.3 Discussion The results presented in this work support the hypothesis (see 1.1) of ˙ U as a second state variable, resulting in a 2-dimensional ring threshold. These thresholds can be seen as separatrices in an U - ˙ U statespace projection which separate the passive pre-spike regime from the active spike dynamic. These separatrices are able to explain the (voltage) threshold variability found in typical intracellular spike recordings, thus making the new threshold concept much more adequate as compared to the 1-dimensional one. We could also show that the algorithms presented here are able to precisely reproduce built-in separatrices from modied leaky integrate-and-re models. Applied to neural intracellular recordings, we additionally could show that neurons have various types of separatrices: vertical, horizontal, slash-type, backslash-type. However, these types are not distinctly segregated, but they seem to form a kind of separatrix-continuum. Furthermore separatrices seem to be a highly sensible indicator not only for the cellular health state, but also for small changes in spike-relevant cell parameters. In this context we could not verify any eect of an A-type potassium channel block. Approaches to attribute certain cell types to certain separatrix types did not succeed. We have not been able to map the separatrix type to the cell type. CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD 54 3.3.1 Model Separatrices The separatrices of the leaky Integrate-and-Fire models precisely reproduce the built-in separatrices. This was to be expected as both the detecting algorithm as well as the predened threshold function work solely in the phenomenological domain. It is, however, a proof for the ability of the algorithm to precisely detect the phenomenological SIPs. The Hodgkin-Huxley separatrices show a more complex situation. Here we see a clear dependence on the starting membrane potential (depolarized, hyperpolarized, rest). This feature is present throughout all Hodgkin-Huxley models tested, but we nd one dierence: while the original Hodgkin-Huxley model has the three oset-related separatrices shifted along their longitudinal axis, all other avors of the Hodgkin-Huxley model have their oset-related separatrices shifted along the spike trajectory (see g. 3.10). Additionally, the found separatrices are unable to reproduce any osetinduced shift along the spike trajectories that occurs in the respective models. In the contrary, they are all grouped more or less together. The exception is the original Hodgkin-Huxley model: Here we see the same shift in the found separatrices that is present in the real separatrices. However, this was to be expected as all separatrices are per denitionem located on the trajectory bundle. If a dierent oset shifts the trajectory bundle perpendicular to the trajectory direction, the separatrices - real as well as found ones - will be shifted as well. It is an interesting nding that Hodgkin-Huxley models seem to exhibit mainly the backslash-type separatrices (although some are admittedly nearly vertical). Although the models tested here were designed with dierent intentions there is none among them that would show a dierent separatrix type. Even further parameter variation did not result in any other separatrix form. It would thus surely be an interesting approach to analyze systematically if and under which conditions Hodgkin-Huxley models are able to switch their separatrix into another type. 3.3.2 Separatrices in General The idea of analyzing neural activity using phase plane projections is not new. It has been used in various mathematical papers (see e.g. [19] and the detailed works of Izhikevich, e.g. [25, 26]) in order to visualize neural behavior in the voltage as well as in the frequency domain using the concepts of limit cycles, attractors, bifurcations and thus even separatrices. However, this approach results from the theoretical analysis of the neuron as a dynamical system. Thus, even if the voltage U is used as one state variable, the other BIBLIOGRAPHY 61 [11] A. N. Burkitt. A review of the integrate-and-re neuron model: I. Homogeneous synaptic input. Biol Cybern , 95(1):119, 2006. [12] A. N. Burkitt. A review of the integrate-and-re neuron model: II. Inhomogeneous synaptic input and network properties. Biol Cybern , 95(2):97112, 2006. [13] B. Cartling. A low-dimensional, time-resolved and adapting model neuron. Int J Neural Syst , 7(3):23746, 1996. [14] P. Cobbett, P. Legendre, and W. T. Mason. Characterization of three types of potassium current in cultured neurones of rat supraoptic nucleus area. J Physiol , 410:44362, 1989. [15] P. Cobbett and W. T. Mason. Whole cell voltage clamp recordings from cultured neurons of the supraoptic area of neonatal rat hypothalamus. Brain Res , 409(1):17580, 1987. [16] C. M. Colbert and D. Johnston. Axonal action-potential initiation and Na+ channel densities in the soma and axon initial segment of subicular pyramidal neurons. Journal of Neuroscience , 16(21):66766686, 1996. [17] A Destexhe. Conductance-based integrate-and-re models. Neural Comput , 9(3):50314, 1997. [18] C. Ebbinghaus. Charakterisierung der neuronalen Feuerschwelle. Master's thesis, Universität Tübingen, November 1997. [19] G. B. Ermentrout. Phase-Plane Analysis of Neural Activity. In The Handbook of Brain Theory and Neural Networks . MIT Press, 1998. ISBN: 0-262-51102-9. [20] E. Ficker and U. Heinemann. Slow and fast transient potassium currents in cultured rat hippocampal cells. J Physiol , 445:43155, 1992. [21] A. Foust, M. Popovic, D. Zecevic, and D. A. McCormick. Action potentials initiate in the axon initial segment and propagate through axon collaterals reliably in cerebellar Purkinje neurons. J Neurosci , 30(20):6891 902, 2010. [22] B. Gutkin and G. B. Ermentrout. Spikes Too Kinky in The Cortex. Nature , 440:9991000, 20 April 2006. BIBLIOGRAPHY 62 [23] H. L. Haas, B. Schaerer, and M. Vosmansky. A simple perfusion chamber for the study of nervous tissue slices in vitro. J Neurosci Methods , 1(4):3235, 1979. [24] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol , 117(4):50044, 1952. [25] E. M. Izhikevich. Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulsecoupled models. IEEE Trans Neural Netw , 10(3):499507, 1999. [26] E. M. Izhikevich. Neuronal excitability, spiking, and bursting. International Journal of Bifurcation and Chaos , 10:11711266, 2000. [27] R. Klee, E. Ficker, and U. Heinemann. Comparison of voltage-dependent potassium currents in rat pyramidal neurons acutely isolated from hippocampal regions CA1 and CA3. J Neurophysiol , 74(5):198295, 1995. [28] C. Koch. The Hodgkin-Huxley model of action potential generation. In The Biophysics of Computation , chapter 6, page 139 . Oxford University Press, 1999. [29] L. Lapicque. Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarization. J. Physiol. Pathol. Gen. , 9:620 635, 1907. Leaky Integrate-And-Fire-Neuron. [30] K. Lucas. On the rate of variation of the exciting current as a factor in electric excitation. J. Physiol. , 36:253274, 1907. [31] Z. F. Mainen, J. Joerges, J. R. Huguenard, and T. J. Sejnowski. A model for spike initiation in neocortical pyramidal neurons. Neuron , 15(6):14271439, 1995. [32] M. Martina, I. Vida, and P. Jonas. Distal initiation and active propagation of action potentials in interneuron dendrites. Science , 287:295300, 2000. [33] D. A. McCormick, Y. Shu, and Y. Yu. Hodgkin And Huxley Model - Still Standing? Nature , 445(10.1038/nature05523):E1E2, 2007. [34] R. Melinek and K. J. Muller. Action potential initiation site depends in neuronal excitation. Journal of neuroscience , 16:25852591, 1996. BIBLIOGRAPHY 63 [35] W. Muller, U. Misgeld, and U. Heinemann. Carbachol eects on hippocampal neurons in vitro: dependence on the rate of rise of carbachol tissue concentration. Exp Brain Res , 72(2):28798, 1988. [36] B. Naundorf, F. Wolf, and M. Volgushev. Unique Features of Action Potential Initiation in Cortical Neurons. Nature , 440:10601063, 20 April 2006. [37] B. Naundorf, F. Wolf, and M. Volgushev. Naundorf et al. Reply. Nature , 445(10.1038/nature05534):E2E3, 2007. [38] H. Querfurth. Action-potential initiation and maintained activity of the isolated frog muscle spindle. Eur J Neurosci , 24(4):114756, 2006. [39] M. J. Shelley and L. Tao. Ecient and accurate time-stepping schemes for integrate-and-re neuronal networks. J Comput Neurosci , 11(2):111 9, 2001. [40] G. Y. Shen, W. R. Chen, J. Midtgaard, G. M. Shepherd, and M. L. Hines. Computational analysis of action potential initiation in mitral cell soma and dendrites based on dual patch recordings. J. Neurophysiol. , 82:30063020, 1999. [41] P. M. Sonner and J. E. Stern. Functional role of A-type potassium currents in rat presympathetic PVN neurones. J Physiol , 582(Pt 3):1219 38, 2007. [42] G. Stuart, N. Spruston, B. Sakmann, and M. Häusser. Action potential initiation and backpropagation in neurons of the mammalian CNS. Trends in Neuroscience , 20(3):125131, 1997. [43] A. Tonnelier, H. Belmabrouk, and D. Martinez. Event-driven simulations of nonlinear integrate-and-re neurons. Neural Comput , 19(12):322638, 2007. [44] A. A. Verveen. On the uctuation of threshold in nerve bre, 1960. [45] X.-J. Wang and G. Buzsáki. Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience , 16(20):64026413, 15 October 1996. [46] Y. Yu, Y. Shu, and D. A. McCormick. Cortical action potential backpropagation explains spike threshold variability and rapid-onset kinetics. J Neurosci , 28(29):726072, 2008. Appendix A Supplements A.1 Abstract It is generally accepted that neural spikes are initiated solely by a voltage threshold mechanism. However, when looking at intracellular voltage recordings, the voltage U exhibits a signicant range of voltage values, thus contradicting the threshold concept as such. From a dynamical systems perspective one clearly would conclude therefrom that the system is insuciently described using U alone as a state variable. Inspired by qualitative reports of neurons that can be driven far beyond their (voltage) threshold when stimulating them slowly enough, we introduced the rst time derivative of the voltage, ˙ U , as a second state variable. Plotting spikes within this two-dimensional state space suggests that they can be locally interpreted as a dynamical system with two attractors: one being the resting potential, the other one being the spike voltage maximum (due to the the Na equilibrium potential). These two attractors can be imagined to be separated by a boundary line, a so-called separatrix. This separatrix would thus be composed of points at which a spike has become unavoidably. In order to verify the separatrix as a 2D ring threshold in U and ˙ U these spike initiation points have to be identied as precise as possible. In physiological data the spike is characterized by a sharp kink at spike onset where it abruptly and vertically emerges from the sub-threshold activity. An algorithm has been developed that - both starting from the sub-threshold as well as from the super-threshold domain - is able to identify these points reliably, precise and robustly. This procedure was veried using Hodgkin-Huxley type and leaky integrateand-re models. Here the real spike initiation points could be determined by 64 APPENDIX A. SUPPLEMENTS 65 successively shortening stimulus length. During analysis it turned out that most Hodgkin-Huxley type models show a very shallow spike onset which in in clear contrast to the sharp onset in cells. We could show that the found spike initiation points are the more distant from the real ones the shallower the spike onset was. On the other hand, if the model showed an abrupt and vertical spike onset real and found spike initiation points were very close together. Because real neurons show an abrupt and vertical spike onset we can thus expect the detected spike initiation points to be identical to the real ones. Both cells and models were then stimulated with ramps of dierent slopes in order to test the threshold dependence on ˙ U . The new algorithm was then used to detect the spike initiation points for the rst spike of each ramp. A curve was tted to these points within the U - ˙ U state space, thus resembling the separatrix of the cell. We could show that all cells exhibit a separatrix. All 4 possible orientation classes were found: vertical (i.e. a pure voltage threshold), horizontal (i.e. a pure ˙ U threshold), slash-type (i.e. from lower left to upper right) and backslash-type (i.e. from lower right to upper left). Hodgkin-Huxley type models showed mainly slash-type separatrices and a vertical one in one case. Leaky integrate-and-re models with predened separatrices were used for verication, and the algorithm was able to reproduce all of these predened separatrices. The neural separatrices showed temporal stability (when cellular state parameters were constant) and independence of positive or negative oset. On the other hand, small changes in cell parameters (membrane time constant, resistance, resting potential) showed clear changes and shifts in the separatrices, thus indicating at them to be a highly sensible indicator of cell state. Pharmacological experiments with A-type channel blockers did not exhibit a systematical eect as - due to the sensitivity addressed above - already the necessary block cocktail changes cell parameters, thus shifting the separatrix. APPENDIX A. SUPPLEMENTS 66 A.2 Zusammenfassung Es ist ein allgemein anerkanntes Kozept, dass neuronale Aktionspotenziale mittels eines einfachen Schwellenmechanismus der Membranspannung U induziert werden. Schaut man sich jedoch die Initiationspunkte von Spikes innerhalb einer Zelle genauer an, so wird eine bemerkenswerte Variabilität der Spannungswerte zu Spikebeginn deutlich, was dem Modell einer Spannungsschwelle jedoch widerspricht. Aus Sicht der mathematischen Konzepte der dynamischen Systeme ist ein solches Phänomen ein Zeichen für einen unterdimensionierten Zustandsraum, in dem (mindestens) eine Zustandsvariable fehlt. Inspiriert durch die immer wieder kolportierten Phänomene, dass Zellen ohne Spike deutlich über ihre Spannungsschwelle depolarisiert werden können, wenn dies nur langsam genug geschieht, wurde die erste Ableitung der Spannung nach der Zeit, ˙ U , als zweite Zustandsvariable eingeführt. Die Darstellung von Spikes in diesem Zustandsraum legt nahe, dass man sie vorübergehend als dynamisches System auassen kann, in dem sich zwei Attraktoren - das Ruhepotential auf der einen Seite, das Spikemaximum (bedingt durch das NaGleichgewichtspotenzial) auf der anderen Seite - benden, die durch eine Separatrix, eine Grenzlinie voneinander getrennt sind. Diese Separatrix müsste demnach aus den Punkten bestehen, an denen im neuronalen Zustandsraum ein Spike unwiderruich initiiert wurde. Für die Verikation der Separatrix als einer 2-dimensionalen Feuerschwelle aus U und ˙ U müssen diese Punkte der Spikeentstehung möglichst präzise bestimmt werden. Phänomenologisch sind sie in den Daten durch einen scharfen Knick gekennzeichnet, in dem der Spikeansatz sich aprupt nahezu senkrecht aus der unterschwelligen Aktivität erhebt. Es wurde ein Algorithmus entwickelt, der sowohl ausgehend von der unterschweligen als auch von der überschwelligen Domäne diesen Spikeinitiationspunkt zuverlässig, präzise und robust nden kann. Dieses Verfahren wurde an Modellen veriziert, in denen zusätzlich die echten Spikeinitiationspunkte durch sukzessive Stimulusverkürzung bestimmt wurden. Dabei wurde deutlich, das die meisten Hodgkin-Huxley-Modelle im Gegensatz zu Zellen einen unphysiologisch achen Spikebeginn besitzen. Während bei Modellen mit schnellem Spikebeginn der gefundene Spikeinitiationspunkt gut mit dem echten übereinstimmt, wird er bei den anderen um so später detektiert, je acher der Spike beginnt. Da die gemessenen Zellen alle über einen aprupten Spikebeginn verfügen, kann davon ausgegangen werden, dass der Algorithmus die Spikeinitiationspunkte präzise detektiert. Zellen und Modelle wurden nun mit verschieden steilen Rampen stimuliert, um die Abhängigkeit der Schwelle von ˙ U zu testen. Für die so APPENDIX A. SUPPLEMENTS 67 induzierten Spikes wurden nun mittels des beschriebenen Algorithmus die Spikeinitiationspunkte bestimmt. An die so gefundenen Punkte wurde im U - ˙ U -Zustandsraum eine Funktion gettet, die die Separatrix für die jeweilige Zelle bzw. das jeweilige Modell darstellt. Es konnte gezeigt werden, dass alle Zellen eine Separatrix besitzen. Es sind alle wurden alle Orientierungen beobachtet: senkrecht (entsprechend einer reinen Spannungsschwelle), waagerecht (entsprechend einer reinen ˙ U - Schwelle) sowie ihre Kombinationen (schräg von links unten nach rechts oben (slash-Typ) als auch von links oben nach rechts unten (backslashTyp)). Hodgkin-Huxley-Modelle zeigten nur eine slash-Typ-Separatrix sowie in einem Fall eine senkrechte. Für Testzwecke wurden spikende Leaky Integrateand-Fire-Modelle mit vordenierten Schwellseparatrices eingesetzt, die von den vorgestellten Algorithmen präzise reproduziert werden konnten. Die Separatrices bei den Zellen zeitlich stabil (bei gleichbleibenden Zellparametern) und unabhängig von einem positiven oder negativen Oset. Andererseits zeigten sich bereits bei kleinen Änderungen der Zellparameter deutliche Änderungen in der Separatrix, was auf eine hohe Sensibilität der Separatrices für den Zellzustand hinweist. Pharmakologische Experimente mit A-Kanal-Blockern zeigten keinen systematischen Eekt, da bereits die Gabe des zur Vermeidung epileptiformer Aktivitäten notwendige Block-Cocktail die Separatrix veränderte, bevor der A-Kanal-Block zugegeben wurde. APPENDIX A. SUPPLEMENTS 68 A.3 Acknowledgements Many people have contributed to this work in a variety of ways. I would like to thank them here, those named below, but also those who I forgot to mention. Andreas Herz gave me the opportunity to pursue these ideas in his Lab. Jan Benda, Daniel Cremers and Martin Stemmler were always available and willing to discuss and criticize approaches and concepts. Laurenz Wiskott helped me to focus thoughts. Petra Prinz and Daniela Eer also discussed ideas and concepts, provided data and especially helped in going through motivational lows. Uwe Heineman supported the experiments both practically as well as theoretically. His experience was extremely valuable to overcome the pits of practical electrophysiology and interpret cellular responses. He also never stopped to support the nishing of this dissertation. Andreas Draguhn and Dietmar Schmitz supported the work with discussions and hints. The experiments were done together with Gunter Kreck with whom I took the rst electrophysiological steps together. Jan Benda, Daniel Cremers and Andreas Draguhn kindly oered their time to thoroughly discuss the rst versions of this dissertation. They made valuable remarks and supported me in nishing the work. Randolf Menzel was immediately supportive when in need of a supervisor. He was interested in the work and supported my attempt to nish it. I thank him very much for being available any time and for his pragmatic calmness. Martin Nawrot was so kind to supervise the last steps of the work. He gave valuable hints on argumentation and structure. Most of all, however, I would like to thank my friend, partner and wife Silke Erdmann for her patience as well as her impatience. She supported me during the lows and corrected me during phases of ignorance. I owe her a lot. And last but not least I would like to thank my daughter Lena Erdmann for her patience and for (mostly) silently accepting that her father could not go swimming because of his dissertation. APPENDIX A. SUPPLEMENTS 69 Contact For further questions you may contact the author at carsten dot erdmann at email dot de .