Manual for the Noisy Spike Generator MATLAB Software: version
Contents
1. The spatiotemporal characteristics can be read in from files These files will be in the subdirectory specified in command line parameter TemporalDirectory and have predefined names for the target neurons the filenames are target_temporal_i where i starts at 0 and for the correlated neurons the filenames are correlated_temporal_z where i starts at 0 The files consist of tab separated floating point numbers The files can be created either using the m file makespatiotemparray or using the supplied Mac OSX application PrepareParams The former stays within MATLAB but has no GUI and the latter need Mac OSX and has a GUI The PrepareParams application provides a GUI for producing the delay characteristics for the target and correlated neurons It produces an array which is 3 by N where N is the number of points The array may then be written either in a form which can be read by the MATLAB function use either the Save Params button or the File Save menu or in a form which can be later incorporated into a MATLAB function use the File Save Matlab menu The user clicks in the view to set a value this value is then linearly interpolated with other values the start and end of the array is set to 0 initially The user selects whether the values being set are for the original first or second derivative using a radio button The user can also select a multiplier for these values However only the sign of the value is eventually2. b spikedist P for Poisson and G for Gaussian distribution for the original signals from each neuron c poissionnumber and poissmeanISI for the Poisson distribution array elements corresponding to Gaussian spike distributions are ignored d gaussmeanITD and gaussstdev for the Gaussian distribution array elements corresponding to Poisson spike distributions are ignored e ttemplateid the number of the template to be used in spike waveform gener ation for this target neuron Must be between 1 and the number of templates Mixing coefficients for the primary signal i origmixcoeffs_orig undifferentiated plain signal strength For extracellu lar recordings this is 0 or very small li origmixcoeffs_d differentiated signal strength The differentiated signals represent the signals from neighbouring neurons contributing to a given electrode The strength of the differentiated signal is user determined A reasonable guide is to make it proportional to 1 d where d is the Euclidean distance between a neuron and the electrode iii origmixcoeffs_dd twice differentiated signal strength This coefficient mod els the effect of glia cells around the electrode The comment above also applies 8 Jittered noise overall size These are used to set the overall size of the total jittered signal If desired the last three can all be set to 1 and the jittered noise mixing coefficients below used on their own It can als
3. significant because the MATLAB programs scale the values It is also possible to read in a set of values using the File Load from file menu Note however that these values cannot then be properly edited This is however useful for displaying parameter files whether created using PrepareParams or makespatiotemparray Parameter file displays may also be saved as pdf files using File Save as PDF image and this can be useful for documentation 4 MATLAB functions We briefly describe below the other MATLAB functions used 4 1 target temporal mac Reads in the delay characteristic for the target neurons The files are in the directory set using TemporalDirectory and the filenames are target_temporal_i where i starts at 0 4 2 correlated_temporal_mac Reads in the delay characteristic for the correlated neurons The files are in the directory set using TemporalDirectory and the filenames are correlated_temporal_i where 7 starts at 0 4 3 uncorrelated_temporal_mac Provides the delay characteristics for the uncorrelated neurons Currently this simply returns a matrix of 1 s 4 4 target_temporal Used when TemporalDatalnFile is false Simply provides an array of the delay coefficients for the target neurons inside the function 4 5 correlated_temporal Used when TemporalDatalnFile is false Simply provides an array of the delay coefficients for the correlated neurons inside the function 4 6 uncorrelated_temporal Ident
4. when the same random number calls are made but currently has a bug DSLength Sets a smoothing parameter used in differentiation simplediff see section 2 Defaults to 60 samples TemporalDataInFile Reports whether the data for delay characteristics is inside a MAT LAB function see section or is to be read from file default TemporalDirectory The name of the directory from which delay characteristics are to be read if TemporalDatalnFile is true Defaults to example inside the current directory ShowSNR A flag to determine if the Signal Correlated Interference Uncorrelated In terference should be displayed after data generation Defaults to 0 don t display Orig_Temporal_Supplied The time data for the target spikes is suppled in the variable following and is not to be read from a file or created by calling target_temporal SameTargetSizes If set then the different target neuron signals will all be stretched or compressed to a specific range currently 0 1 0 1 Target1 Weights The strengths for the components of the target 1 neuron are supplied in the following variable this is a 3 tuple being the strength of the original signal of the differentiated signal and of the twice differentiated signal 2 3 The target structure returned The target structure does not need to be accessed to use the software However it can provide useful information The target structure returned is a 1 x N_Targets array of structures
5. Manual for the Noisy Spike Generator MATLAB Software version 1 1 released 16 June 2006 Leslie S Smith and Nhamoinesu Mtetwa Deprtment of Computing Science and Mathematics University of Stirling Stirling FK9 4LA Scotland UK contact lssQ cs stir ac uk 1 Introduction This tool is used to generate noisy spike trains perhaps better described as spike trains with interference to simulate the kind of signals that an extracellular electrode such as those on a multi electrode array record from a neural culture This document is the user manual for the software a separate document describing the analysis underpinning this is also available 2 The tool is intended for use in testing spike detection and spike sorting algorithms it provides a method of generating realistic spike trains for which the ground truth is known It can also be used simply to generate background noise or interference of the sort that might be encountered by an extracellular electrode The software generates signals from a single electrode It can allow for e a number of neurons at different distances from the electrode and with different neuron electrode geometries some generating correlated spikes and some generating uncorrelated spikes e the electrode being partially covered by glial cells The signal that a real electrode records is a summation of intracellular spikes from many neurons arising from ionic flow over the surface of these membranes This sig
6. allow the replication of the effect of the spike over the extent of the spiking surface of the neuron To achieve this we consider only the delay and the relative strength of the signal for that delay The actual structure of the neuron is assumed to be captured purely by the strength of the delayed signal see 2 The delay system uses two values set either in the code or in the call to generatenoisyspikes see section 2 1 In the code the variables are t_delta_integrate the timestep used in the delay integration This needs to be quite small default is 30useconds because the original signal rises very rapidly and thus contains harmonics up to a high frequency Logically t_delta_integrate should be less than half the maximal frequency of interest n_delta_integrate the number of timesteps used This defaults to 60 11 The result is that the default implies a maximal spike generation duration of the product of these two that is 1 8 milliseconds Three arrays are used to determine the actual weights to be applied at each of the delay instants orig temporal array determining the relative weights for each delay for the target signals corr_temporal array determining the relative weights for each delay for the correlated signals u_temporal array determining the relative weights for each delay for the uncorrelated signals Each of these arrays is No of neurons by 3 by n_delta_integrate they can be different for each neuron and
7. ar signal to electrode for example a layer of glial cells covering electrodes at the bottom a an MEA culture dish resulting in further differentiation What is actually recorded at the electrode is the sum of all of these weighted by their size and integrated over the neural surface for many neurons Importantly the signal generated by a spiking neuron does not all occur at the same time In a real neuron the spike is initiated at the axon hillock then travels down the probably branching axon The transmission speed depends on a number of factors axon diameter and myelinisation in particular What matters here is that spikes produced as a result of a single axon hillock action potential are present in the neuron for a relatively long time compared to the duration of the spike itself with the default parameters this time can be up to 1 8 milliseconds but this can easily be altered Further particularly for neurons which are reasonably close to the electrode some parts of the structure of the spiking part of the neuron are likely to be much closer to the electrode than others As a result the shape of a signal received for a single spike from a single neuron is likely to be characteristic of that neuron permitting spike sorting to occur We model this by adding up weighted delayed versions of the signal that might be expected at the electrode from a single intracellular spike at the axon hillock In fact we compute weighted summed versions
8. arameters using the MATLAB varargin facility thus to set the duration of the simulation one puts Duration 0 3 in the command line The command line parameters are Duration This is the duration of the spike train in seconds It defaults to 0 1 seconds SampleRate The number of samples second It defaults to 100000 samples second N_Targets The number of target neurons Defaults to 2 N_Jitter The number of jittered spike trains Defaults to 7 N_Uncorr The number of uncorrelated independent spike trains Defaults to 15 RefractoryPeriod The neuron refractory period in seconds Defaults to 0 001 seconds T_Delta_Integrate The single delay used in summing over the time that a spike takes throughout the neuron spiking surface Defaults to 30u seconds Needs to be quite short for Nyquist reasons N_Delta Integrate Number of delay times Defaults to 60 which gives a default spike duration of 1 8ms ReuseTargets Reuse the target structure i e times and all target characteristics al lows different sets of parameters to be used with the same target spiking times useful for testing out the effects of different levels and forms of noise ReuseTargetTimes Reuse the target spiking times i e just the times of the target neuron spikes Can be used as above and also to experiment with the effects of different delay characteristics ReuseRNGState Reuse the random number generator state Should lead to exactly the same behaviour
9. d the first spike template sample with the value nearest to the current value is used as the start of the next spike There are implicit assumptions in this i spikes are spike shaped that is the consist of an upswing then a downswing and probably a post spike hyperpolarization and ii the refractory period is longer than the spike upswing Both of these are normally true for neurons but are not necessarily true for the values supplied to a tool like this one One or more templates may be supplied Two very similar ones generated using a MATLAB Hodgkin Huxley simulator 3 are supplied spike_In csv and spike_2n csv The format for these files is simply a comma separated set of values Each line contains a time and a membrane voltage and possibly other values which are ignored The times are in milliseconds and the voltages are in millivolts Note that the times need not start at 0 but the template will be normalised so that it does start at 0 The times do not need to be equally spaced out the system will interpolate using cubic interpolation so that a value for the template at each sample time is produced The start and end values for the membrane voltages should be the same or very similar In addition we have supplied another one based on 1 in a file naundorf_vivo csv 6 List of internal alterable values Virtually anything that can be set is set near the start of gennoisyspikes 6 1 Temporal Information The aim is to
10. e number of the template to be used in spike waveform gen eration for this uncorrelated neuron Must be between 1 and the number of templates ee 14 h Uncorrelated spikes mixing coefficients set the level of the signal from each uncorrelatied neuron i umixcoeffs_orig for intracellular signal ii umixcoeffs_d for 1st differential iii umixcoeffs_d for 2nd differential 12 noise_snr noise level in dB for the awgn MATLAB add white Gaussian noise function 13 fminval final minimum value for the output signal 14 fmazxval final maximum value for the output signal 7 Some examples File generatenoisysamples m provides an example which when called with no parameters produces a signal output lasting 0 5 seconds with two target outputs 7 jittered spike trains and 15 independent trains or whatever the defaults have been re set to inside gennoisyspikes m When called signals1 target1 r1 generatenoisysamples N_targets 4 it produces a signal with four target outputs and the same number of jittered and inde pendent spike trains It is possible to produce purely noisy signals with no targets or correlated signals simply set the number of targets and correlated outputs to 0 In addition the subdirectory Extras contains a number of m files which were used for as sessing the KwikKlusters see http klustakwik sourceforge net and wave_clus see http www vis caltech edu rodri Wave_clus Wa
11. effs_d for 1st differential iii jmixcoeffs_dd for 2nd differential 10 Uncorrelated spike noise overall size The last three can be used to set the overall size of the uncorrelated noise signal Alternatively if desired the last three can all be set to 1 and the Uncorrelated spikes mixing coefficients used to adjust signal levels It can also be useful to use ucoveralllevel to adjust the overall size of the uncorrelated interference a ucoveralllevel a scalar used to adjust the overall size of the uncorrelated inter ference b c d e 11 All arrays below should be the same length as u_temporal If n_uncorr is greater than this the values used will cycle through these over and over again n_uncorr number of uncorrelated spike trains can also be set in call u_overallsize_orig overall size of the original unjittered signal u_overallsize_d overall size of the differentiated unjittered signal u_overallsize_dd overall size of the twice differentiated unjittered signal a uncorr_temporal weights for the delayed uncorrelated neurons see orig_temporal above o Seo RS NR EE ES u_spikedist distribution type for these spikes u_poissonnumber number for use when generating Poisson distribution eS u_poissonmeanISI mean ISI for each poisson distribution u_gaussianmeanITD mean interspike time difference should be ISI u_gaussstdev STD thereof fo a u_templateid th
12. erent noise being added Calling the function using signals2 target2 ri generatenoisysamples ReuseRNGstate ri Duration 0 2 SampleRate 100000 should result in exactly the same times and the same random numbers being generated If no changes are made to the internal variables this should generate exactly the same signal Normally of course one makes some changes for example to the mixing coefficients so that one can control exactly what is generated Bug currently the signal generated is not identical The function starts by reading in the characteristic delays These may be read from a file using the functions target_temporal_mac for the target neurons correlated_temporal_mac for the correlated neurons and uncorrelated_temporal_mac for the uncorrelated neurons or incorporated inside functions target_temporal for the target neurons correlated_temporal for the correlated neurons and uncorrelated_temporal for the uncorrelated neurons details of these functions a provided in section 4 below Which happens depends on the command line argument TemporalDatalnFile The function then initialises a host of internal variables see section 6 2 for details After all the parameters are initialised a number of spike templates are loaded from files Next spike times are created for each target neuron using either a Gaussian or Poisson distribution The spike times are stored in the target structure The number of
13. erentiated twice using the function simplediff to produce jittered versions of once and twice differentiated spikes again further differentiation could easily be added and the effect of the geometry of these neurons and the electrode implemented using spatiotransform The differentiated signals are also normalised using the function setlimits The number of jittered neurons used can be set by the user the amount of jitter can be different for each 2 5 Uncorrelated spikes This is intended for modelling the effect of nearby neurons whose spike times are not correlated with the original spike times Gaussian or Poisson distributed uncorrelated spike times are created using the functions genspikesgaussion or genspikespoisson and the uncorrelated spike times are stored in the vector u_stimes These uncorrelated spike times are then used to create a membrane voltage signal based on a spike template see section 5 with uncorrelated spikes using gensampledspikes As before two normalised differentiated versions of this signal are also created and the effects of the delays in the neuron applied The number of different neurons used can be set by the user 2 6 Correlated spike noise Correlated spike noise is created by linearly mixing the jittered spike train once differenti ated spike train and the twice differentiated jittered spike train created above The mixing is done using coefficients in vectors namely jmizxcoeffs_orig jmixcoeffs_d and jm
14. for the original 1st derivative and 2nd derivative Each value is the weight for one neuron for one of the three of original 1st and 2nd derivative and for one time delay One vector should be supplied for each target neuron For the correlated and for the uncorrelated neurons the weights to be used will simply go round and round the array supplied so that there may be fewer vectors in this array than actual uncorrelated neurons The values are read in either from a file or from numbers inside MATLAB m files see section 4 6 2 Other parameters The following parameters are set in generatenoisysamples m 1 sample_rate the sampling rate for the signals can also be set in the call 2 templatenames filenames for file with template for spike for primary neuron Cur rently these file names must all be of the same length 3 duration length of the final signal in seconds can also be set in the call 4 n_targets number of target neurons can also be set in the call 5 dslength used in smoothing during differentiation It is a number of sample intervals It can also be set in the call 6 refractotoryperiod the default refractory period It can be set in the call 7 The arrays below must all be at least n_targets long a orig_temporal a set of vectors detailing the relative effects of the delays inside the target neurons Each vector must be N_Delta_Integrate long and there must be at least n_targets of them 12
15. ical to uncorrelated_temporal_mac 4 7 makespatiotemparray This function takes a number of parameters the first is the length L of the spatiotemporal array What it does is to take a set of times and values for the strengths for the original signal and its first and second derivatives and using linear interpolation produce a 3 by L array Where extrapolation would be necessary the values are set to 0 4 8 genspikespoisson This generates a Poisson distributed spike train Both A and the mean inter spike interval are parameters It is also possible to set the minimum inter spike interval using varargin The keyword is MinISI 4 9 genspikesgaussian This generates a Gaussian distributed spike train Both the mean inter spike interval and the standard deviation can be set The minimum inter spike interval is set as above 4 10 gensampledspikes This function takes in 1 a 1 d array of spike times in seconds 2 an array describing the intracellular spike template This is a csv comma separated values file with each line describing one spike point there may be many values per line but only the first two are used These are the time in milliseconds and the voltage in millivolts It produces from this a 1 d array of the membrane voltage at sample times with spikes placed at times specified in the input spike times 1 d array The template spike is resampled to the required sample rate before being placed at spike position accord
16. ing to the predetermined spike times 3 a variable number of other arguments MinVal MaxVal The minimum and maximum value of the signal to be produced The signal will be linearly adjusted to be between these values EndTime The duration of the signal to be produced 4 11 simplediff This function performs a simple differential operation It smooths the input signal first using a hamming window of a given length The function returns a vector same length as input 4 12 spatiotransform Essentially convolves the delay coefficients with the signal A little complexity arises be cause the inter sample times differ so that the delay coefficients need to be interpolated first 4 13 probjitter This function produces a new list of spike times from the old list Each output spike is produced from each input spike with probability probspike at a time determined from a gaussian distribution with mean 0 and standard deviation jitterstd 4 14 setlimits This function linearly scales the input values to be within the specified maximum and minimum values The function basically normalises or stretches the spikes 10 5 Spike Templates The spike times are turned into intracellular voltages using spike templates This allows different neurons to have different spike shapes Each spike time is taken to be the start of a spike and the appropriate spike template is added in Where spike templates would overlap they are not simply added Instea
17. ixcoeffs_dd In addition one can vary the overall size of each of these signals by using jitteroverall size_orig jitteroverallsize_d and jitteroverallsize_dd allowing the original differentiated and twice differentiated signals to be varied in size without adjusting each vector 2 7 Uncorrelated spike noise Uncorrelated spike noise is created by linearly mixing the uncorrelated spike train once dif ferentiated and the twice differentiated uncorrelated spike trains created above The mixing is done using coefficients in vectors namely umixcoeffs_orig umixcoeffs_d and umixcoeffs_dd In addition one can vary the overall size of each of these signals by using u_overallsize_orig u_overallsize_d and u_overallsize_dd allowing the original differentiated and twice differ entiated signals to be varied in size without adjusting each vector 2 8 Final signal The final signal is created in four stages 1 the original signals are linearly mixed using coefficients described in section 6 2 with their once and twice differentiated versions 2 the result of this is linearly mixed with correlated and uncorrelated spike noise signals generated as above 3 the result of the second mixture is corrupted by some additive Gaussian white noise of a chosen strength using the function agwn 4 the signal thus generated is linearly scaled to be between the final output values fminval and fmaczval 3 Generating the spatiotemporal characteristics
18. mes of the spike templates so that the actual peak will be some little time later It also returns the clean signal from each target neuron as would be received if there was no noise and no other target neurons This structure can be used to generate new data sets which have the same primary spikes as a previous data set but different noise levels etc see section 2 3 for details of the structure 3 a structure used by the MATLAB random number generator Returning it allows the same random numbers to be generated the next time the function is called if this is desired 2 1 generatenoisysamples The function may be called as follows signals target ri generatenoisysamples Duration 0 2 SampleRate 100000 and this returns the samples as a 1 by N array where N 0 2 x 100000 20000 in signals Information about the target signals is returned in target see section 2 3 r1 returns the state of the random number generator The number of each type of neuron and all the other parameters are then taken from the m file itself The intracellular spike shapes are taken from files see section 5 and the information about the spatiotemporal characteristics are also taken from files see section 4 and 3 Calling the function a second time using signals1 targeti r1 generatenoisysamples ReuseTargets target Duration 0 2 SampleRate 100000 results in the original spike times being kept but diff
19. nal is thus the result of the spatiotemporal distribution of the spikes on the spiking surface of the neuron and of the path that the signal takes to the electrode For the purposes of this tool neurons in the culture are divided into target neurons which are those that one might be hoping to detect and sort correlated neurons whose spikes are correlated with one of the target neurons but which are to be considered as correlated interference and uncorrelated neurons which are intended to provide uncorrelated interference The number of each type can be set either in the program or from the invoking MATLAB command Since the electrode is extracellular we assume that the signal that it picks from any patch of the surface of a neuron will be a summation of a number of forms of transfor mation discussed in detail in the report document 2 Signals crossing the neuron s membrane undergo differentiation due to the capacitance of the membrane and also due to the mechanism of spike re generation Further differentiation can occur because of the nature of signal transfer from the extracellular electrolyte to the electrode particularly if the electrode is covered by insulating glial cells But it is also the case that some signal might arrive directly having undergone a primarily resistive path from intracellular signal to electrode Additionally there is the possibility that there might be other glial cells in the path from intracellul
20. o be useful to use jitteroveralllevel as a single value to adjust the overall jittered interference size a jitteroveralllevel overall size of the jittered noise a simple scalar multiplier useful for scaling the values b n_jitter number of jittered versions of the original spike train can also be set in call c jitteroverallsize_orig overall size of the original jittered signal d jitteroverallsize_d overall size the differentiated jittered signal strength e jitteroverallsize_dd overall size the twice differentiated jittered signal strength 9 The arrays below should be the same length as the number of vectors in corr_temporal If this is less than n_jitter the values used will cycle through these over and over again a corr_temporal delay weight values for the jittered neurons see orig_temporal above b jprobspikes probability that a spike produced actually results in an output spike 13 c jjstd standard deviation of the jitter mean 0 d jitter_orig the number of the target neuron on which this jittered neuron s spikes are based Must be between 1 and the number of target neurons e j templateid the number of the template to be used in spike waveform genera tion for this jittered neuron Must be between 1 and the number of templates Jittered noise mixing coefficients set the level of the signal from each jittered neuron i jmixcoeffs_orig plain intercellular signal ii jmiaxco
21. of the intracellular potential then use this and its derivatives discussed above relying on the linearity of summation and differentiation Lastly we can add some Gaussian noise to mimic the effect of thermal and amplifier noise As the program stands adjusting the values of the parameters can be achieved partly from the command line but may also require going into the program and altering values inside reasonably well commented MATLAB m files The shape of the intracellular spike is held in a file which can be written using HHSim 3 The characteristics of the effect of the neuron surface electrode geometry are held in a file for each neuron This file can be produced either using the GUI of the supplied Mac OSX application PrepareParams or using the m file spatiotemporal m see section 3 or simply edited by hand Even although the MATLAB m files are quite well commented there really ought to be a better user interface One day 2 The program The main function m file is generatenoisysamples This function calls a number of func tions and reads some files to produce a noisy spike train generatenoisysamples returns three variables namely 1 the noisy signal a 1 dimensional array 2 a structure containing amongst other things the actual official spike times These include both the peaks of the intracellular spikes and the spike times used to gen erate the spike train Note that the spike times are the starting ti
22. struct array in MATLAB terminology There is one structure per target neuron The structure fields are targettimes The times of the start of each spike sampletargets The intracellular spiking signal prior to integration over the spiking sur face of the neuron stsamples The intracellular spiking signal after integration over the spiking surface of the neuron actualpeaks The actual peaks in the integrated intracellular spike signal ssamples_In Same as stsamples but normalised to be between the minimum and maxi mum value 0 5 and 0 5 ssamples_1dn Differentiated signal normalised to be between the minimum and maxi mum value 0 5 and 0 5 ssamples_1ddn Double differentiated signal normalised to be between the minimum and maximum value 0 5 and 0 5 final The signal received by the electrode from this neuron without any noise added 2 4 Jittered spikes This is intended for modelling the effect of nearby neurons whose spike times are correlated with but not identical to one of the original spike times The original spike times are jittered using probjitter which randomly shifts the position of each spike by a few samples to create a vector jstimes of jittered spike times The jittered spike times are used to produce another membrane voltage signal based on a spike template see section 5 with jittered spikes using the function gensampledspikes As with the target neurons the signal with jittered spikes is also diff
23. target spike trains is set using N_Targets an optional argument to the function see below this defaults to two unless the value has been reset in the m file file note that all the defaults below can easily be altered Gaussian or Poisson distributed uncorrelated spike times are created using the functions genspikesgaussion or genspikespoisson The next function to be called is gensampledspikes This generates sampled spikes from the spike times by resampling the appropriate template spike see section 5 to make sure that the spikes are evenly sampled The resulting signal is used to produce two other signals once differentiated and twice differentiated by calling the function simplediff which performs a differentiation operation on the signal Differentiation also includes some smoothing to remove artefacts caused by inaccuracies in the original spike signal Note that further differentiation can be included by adding further calls to this function For both the original signal and its two derivatives the function spatiotransform is then called and this adds in weighted delayed versions of the signal emulating the effect of the time dispersion of the spike Lastly these three signals are normalised to between 0 5 and 0 5 using the function setlimits A similar approach is taken to generating both the correlated and uncorrelated inter ference signals 2 2 Command line arguments generatenoisysamples can take a number of command line p
24. ve_clus_home htm spike sorting pack ages The m files in this directory have been commented to explain their function they are probably not directly useful for further research but may be a template And finally These MATLAB m files are not guaranteed in any way or sense whatsoever Nonetheless we do attempt to make sure that they are bug free If you find errors please tell us about them In addition if you add new routines which might prove useful to others tell us about them as well and we ll put in links to them 15 References 1 B Naundorf F Wolf and M Volgushev Unique features of action potential initiation in cortical neurons Nature 440 7087 1060 1063 2006 2 L S Smith and N Mtetwa Generating realistic extracellular noisy spike trains for testing detection and sorting algorithms in preparation 2006 3 D S Touretzky M V Albert N D Daw and A Ladsariya HHsim Graphical Hodgkin Huxley simulator at http www cs cmu edu dst HHsim 2004 16
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