Spike separation based on symmetries analysis in phase space

The present study introduces an approach for automatic classification of extracellularly recorded action potentials of neurons based on geometrical approach. Neuronal spikes are considered as geometrical objects, namely trajectories in phase space. It is shown that for spikes, generated by the same...

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Дата:	2003
Автори:	Polyarush, A.I., Makarenko, A.S., Tetko, I.V.
Формат:	Стаття
Мова:	English
Опубліковано:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2003
Назва видання:	Системні дослідження та інформаційні технології
Теми:	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/50276
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Spike separation based on symmetries analysis in phase space / A.I. Polyarush, A.S. Makarenko, I.V. Tetko // Систем. дослідж. та інформ. технології. — 2003. — № 2. — С. 124-135. — Бібліогр.: 9 назв. — англ.

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spelling	irk-123456789-502762013-10-09T03:07:06Z Spike separation based on symmetries analysis in phase space Polyarush, A.I. Makarenko, A.S. Tetko, I.V. Нові методи в системному аналізі, інформатиці та теорії прийняття рішень The present study introduces an approach for automatic classification of extracellularly recorded action potentials of neurons based on geometrical approach. Neuronal spikes are considered as geometrical objects, namely trajectories in phase space. It is shown that for spikes, generated by the same neuron, it is possible to find such a symmetry transformation under which their trajectories are invariant in phase space. On the other hand, the phase trajectories of spikes generated by other neurons change significantly under the action of that transformation. Thus, it is possible to define a special symmetry transformation that only typifies the spikes of the given neuron. The proposed algorithm is explained and an overview of the mathematical background is given. The method was tested on simulated data and showed good results in real experiments. Запропоновано підхід до автоматичної класифікації внутрішньокліткових потенціалів нейронів, заснований на геометричному підході. Нейронні спайки (викиди) розглядаються як геометричні об’єкти, а саме як траєкторії у фазовому просторі. Показано, що для спайків, згенерованих одним нейроном, можна знайти такі перетворення симетрії, под дією яких ці треєкторії інваріантні у фазовому просторі. З іншого боку, фазові траєкторії спайків, сгенерованих іншими нейронами, змінюються значною мырою під дією перетворень. Таким чином, можна ввести спеціальні перетворення сіметрії, які відповідають конкретному нейрону. Описано запропонований алгоритм та наведено огляд математичних основ методу. Метод тестовано за спеціальними даними, отримані позитивні результати. 2003 Article Spike separation based on symmetries analysis in phase space / A.I. Polyarush, A.S. Makarenko, I.V. Tetko // Систем. дослідж. та інформ. технології. — 2003. — № 2. — С. 124-135. — Бібліогр.: 9 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/50276 681.142.36 : 153.7 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
spellingShingle	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Polyarush, A.I. Makarenko, A.S. Tetko, I.V. Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології
description	The present study introduces an approach for automatic classification of extracellularly recorded action potentials of neurons based on geometrical approach. Neuronal spikes are considered as geometrical objects, namely trajectories in phase space. It is shown that for spikes, generated by the same neuron, it is possible to find such a symmetry transformation under which their trajectories are invariant in phase space. On the other hand, the phase trajectories of spikes generated by other neurons change significantly under the action of that transformation. Thus, it is possible to define a special symmetry transformation that only typifies the spikes of the given neuron. The proposed algorithm is explained and an overview of the mathematical background is given. The method was tested on simulated data and showed good results in real experiments.
format	Article
author	Polyarush, A.I. Makarenko, A.S. Tetko, I.V.
author_facet	Polyarush, A.I. Makarenko, A.S. Tetko, I.V.
author_sort	Polyarush, A.I.
title	Spike separation based on symmetries analysis in phase space
title_short	Spike separation based on symmetries analysis in phase space
title_full	Spike separation based on symmetries analysis in phase space
title_fullStr	Spike separation based on symmetries analysis in phase space
title_full_unstemmed	Spike separation based on symmetries analysis in phase space
title_sort	spike separation based on symmetries analysis in phase space
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
publishDate	2003
topic_facet	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
url	http://dspace.nbuv.gov.ua/handle/123456789/50276
citation_txt	Spike separation based on symmetries analysis in phase space / A.I. Polyarush, A.S. Makarenko, I.V. Tetko // Систем. дослідж. та інформ. технології. — 2003. — № 2. — С. 124-135. — Бібліогр.: 9 назв. — англ.
series	Системні дослідження та інформаційні технології
work_keys_str_mv	AT polyarushai spikeseparationbasedonsymmetriesanalysisinphasespace AT makarenkoas spikeseparationbasedonsymmetriesanalysisinphasespace AT tetkoiv spikeseparationbasedonsymmetriesanalysisinphasespace
first_indexed	2025-07-04T11:51:56Z
last_indexed	2025-07-04T11:51:56Z
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fulltext	© A.I. Polyarush, A.S. Makarenko, I.V. Tetko, 2003 124 ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 UDC 681.142.36 : 153.7 SPIKE SEPARATION BASED ON SIMMETRIES ANALYSIS IN PHASE SPACE A.I. POLYARUSH, A.S. MAKARENKO, I.V. TETKO The present study introduces an approach for automatic classification of extracellularly recorded action potentials of neurons based on geometrical approach. Neuronal spikes are considered as geometrical objects, namely trajectories in phase space. It is shown that for spikes, generated by the same neuron, it is possible to find such symmetry transformation under which their trajectories are invariant in phase space. On the other hand, the phase trajectories of spikes generated by other neurons change significantly under action of that transformation. Thus it is possible to define a special symmetry transformation that only typifies the spikes of the given neuron. The proposed algorithm is explained and an overview of the mathematical background is given. The method was tested on simulated data and showed good results in real experiments. INTRODUCTION Analysis of populations of neurons represents an important step for better under- standing of brain functioning. The present theories of brain functioning stress ac- cent on the neuronal interactions in term of syncronized firing across many neu- rons, or spatio-temporal interactions and presence of specific patterns. Many such neuronal interactions cannot be observed with recording from single neuron. Thus, analysis of neuronal populations does not simply provide an additive scheme for increase experimental data but make possible to detect some features of brain functioning that could never be observed with recording of only single neuron. The basic hypothesis used to detect and separate action potential of neurons assumes that spikes generated by the same neurons have similar shapes and these shapes are unique and conservative for each recorded neuron. The shape of neuron spikes detected by the electrode depends on the distance, relative position, properties of the media, e.g., presence and distribution of glia cells, between electrode and neurons. The shape also depends on resistance of electrode and neuron. Thus, even if two neurons are identical, their action potential detected at the microelectrode can be different and, thus, spikes from such neurons could be separated. The number of different methods used in the neurophysiology for spike sorting dramatically increases. One of the most popular methods is template matching. These techniques use templates that represent some typical waveform shapes of neurons in time domain. A classification of a candidate spike is done by its comparison to all templates and selecting the best matching one. The recent trends include appearance of more computationally expensive methods, such as neural networks wavelet transforms. Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології, 2003, № 2 125 There are many basic problems should be solved for successful spike sorting. The first basic question concerns the number of different types of neurons that should be separated in the experimental data. The usual practice is to use a «supervisor», i.e. the experimentator, who can provide a preliminary classification of data, e.g. selection of template spikes, based on his experience and knowledge. This is, for example, a feature of Multi Spike Detector (MSD) software (Alpha Omega Ltd., Nazareth) that is based on the template matching algorithm of. However, even if number of classes is identified, the separation of spikes remains very difficult problem due to extracellular and intracellular noise that can disturb the form of the action potential. The extracelular noise is usually taken into account by the most of models as an additive noise. The intracellular noise that can produce variation in the spike waveform is more difficult to account for. Recently we have proposed a new method for spike sorting that consider the problem of spike sorting in phase space and describe the spike waveform as an ordinary differential equation with perturbation [1]. This approach made possible to account for both extracellular and intracellular noise. The differential equation describing the activity of a neuron was supposed to have a limit trajectory in phase space, and noise was treated as deviation of the signal from that trajectory. Current study provides further development of this idea. As contrasted to [1], we proposed a numerical method that takes into account that the variety of spike waveforms generated by the same neuron cannot be only explained by influence of both kinds of noise on some «typical» for a given neuron impulse, but is should be considered as a whole. Therefore, the new theoretical background uses a wider class of differential equations, not necessarily describing self-oscillating systems. Another principal difference is that the activity of a neuron is described by a symmetry transformation in phase space. Criterion of classification is the steadiness of the portrait in phase space of a spike against the transformation that corresponds to the given neuron. A computational method for modeling transformations is introduced and tested. 1. MATHEMATICAL STATEMENT OF THE PROBLEM Let us consider a microelectrode signal )()()( 0 ttxtx ξ+= that is observed at dis- crete times ...,1,0=t . Here )(tξ is a sequence of independent identically distrib- uted random variables with zero mean and finite variance ( ∞<2 ξσ ). The signal )(tx is characterized by the occurrence of repeated intervals with amplitudes sig- nificantly exceeding the variance of )(tξ . These intervals are assumed to be the occurrences of neuronal discharges, i.e. the spikes. Let us suppose that every observed spike )(txi of the i -th neuron is a solu- tion of an ordinary differential equation with perturbation ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 2 2 2 3 3 ,,,,, dt xd dt dxxtF dt xd dt dxxf dt xd i , (1) A.I. Polyarush, A.S. Makarenko, I.V. Tetko ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 126 where )(⋅F is a perturbation function. The perturbation function ),...,( txF , bounded by a small value, is a random process with zero mean and small correla- tion time T<<*τ . The solution of the equation ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 2 3 3 ,, dt xd dt dxxf dt xd i (2) describes a self-oscillating system. The duration of spikes is limited, so we can suppose that )(tx is defined on some interval ( )ba; . In practice parameters a and b are chosen manually by an expert. Set of differential equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎪ ⎨ ⎧ = = = ,,, , , 210 2 2 1 1 0 xxxf dt dx x dt dx x dt dx i (3) where )(kx is the k -th derivative of )(tx is equivalent to (2). If function if is defined in open domain 3RD ⊂ then any solution x(t) of (3) induces phase trajectory ( ) ( ) ( ) ( ) ( ) ( )( )TtxtxtxtX 210 ,,)( = . In this case the trajec- tory is also an integral curve for (3). The theorem about existing of a solution for an ordinary differential equation applies that for an arbitrary point Dp ∈ exists such integral curve ( )tX p that ( ) ptX p =0 for some t0 and ( )tX p is a solution of (3). The equation (3) induces local one-parameter group G on D [2]. Let us de- fine some action Gg ∈τ on D . Consider integral curve ( )tX p that passes p when 0tt = . Then ( )0: tXpg p +ττ (4) i.e., ( )pgτ describes the state of system (3) at time moment 0t+τ under initial conditions p. If the action of G on D is known we can introduce a criterion that allows to determine whether an arbitrary function x(t), defined on interval ( )ba; , is a solu- tion of (3). That is, ( ) ( ) ( ) ( ) ( ) ( )( )TtxtxtxtX 210 ,,)( = is an integral curve for (3) if and only if ( )( ) ( )tXtXg =−ττ (5) for any ( )bat ;∈ and any permissible ( )εετ +−∈ ; . Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології, 2003, № 2 127 Let us consider set Φ of all curves ( ) Dba →;:ϕ . (6) A natural distance function on Φ can be given as ( ) ( )( ) ( ) ( )( )∫ −=⋅⋅ b a dttt 2 2121 , ϕϕϕϕρ . (7) The action of G on Φ can be defined as: ( )( ) ( )( )τϕϕ ττ −=⋅ tgtg )( . (8) If ( )tϕ is a solution of (3) if and only if ( )( ) ( )⋅=⋅ ϕϕτg for any permissible ( )εετ +−∈ ; . This statement is a formalization of (5) in terms of functions. Let us introduce a deviation function R→Φ∆ :τ , (9) ( ) ( )( ) ( )( )⋅⋅⋅∆ ϕϕρϕ ττ ,: g . (10) In other words, τ∆ is a numerical criterion that shows how accurately an ar- bitrary phase trajectory ( )tϕ can be described by differential equation (2). ( )tϕ is a solution of (3) if and only if ( )( ) 0=⋅∆ ϕτ for any permissible τ . Consider an arbitrary spike )(tx that solves system (1) and its phase trajec- tory ( )tϕ . Point ( )0tp ϕ= gives an initial condition for system (1). If the pertur- bation ),,,( 2 2 dt xd dt dxxtF is small enough then the phase trajectory ( )tϕ of )(tx will stay close to phase trajectory ( )tX p of some solution )(tx p of undisturbed system (3) for ( )εε +−∈ 00 ; ttt , where ε is small enough (Fig.1). X(t) )()( 0 xtXpg p +=τ)( 0tXp p=D Fig. 1. Vector field on D and its integral curves that correspond to solutions of equation (3). The action of tg on point p is shown A.I. Polyarush, A.S. Makarenko, I.V. Tetko ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 128 It is true for any ( )bat ;0 ∈ . There- fore the phase portrait ( )tϕ of the ob- served spike will be locally steady against the transformation with group G for every ( )bat ;0 ∈ . Deviation function (10) makes it possible to obtain a numerical criterion for the whole interval ( )ba; . The greater is the influence of the perturbation F , the greater is the value of τ∆ (Fig.2). So, one of the differences of the method presented from our previous work [1, 3–6] is that we do not put into accordance to the undisturbed system the only limit phase trajectory, but character- ize it by group of transformations G , which describes the whole set of solu- tions of (2). We also have developed some ideas on symmetry application [8, 9]. The other difference is that instead of calculating the distance between the phase trajectory ( )tϕ of an observed spike and the limit trajectory chosen as a sample for a given neuron we analyze the steadiness of ( )tϕ against the action of group G . The set of integral curves that correspond to all solutions of equation (3) defines a vector field V on D . Sup- pose we know how group G acts on subset D of phase space. Then we can esti- mate how well the following spike classification criterion can be. 2. DESCRIPTION OF ALGORITHM STEPS The algorithm to separate neuronal spikes several intermediate steps: 1. Spike detection from the noisy signal; 2. Calculation of distances between the phase trajectories of the detected spikes; 3. Detection of spikes that hypothetically belong to the same neuron; 4. Building a numerical model of transformation group that correspond to every neuron observed; 5. Classification of spikes. The three first steps were performed similar to our previous analysis and are described in details in elsewhere [1, 4, 5]. In this article we only briefly summa- rize the main implementation details and parameters of calculations of these steps that are important to reproduce our work. The vector field approach (steps 4 – 5) is described in details and main differences between old and new algorithm are emphasized. The first 4 steps corresponded to the training phase on which the number of spike classes was estimated and the vector field for each class was constructed. In order to perform this analysis few tens of spike occurrences, usually correspond- yx n( t1) n( t2) t1 t2 Fig. 2. The deviation of spike y(t) relative to spike x(t). Deviation vector n(t) at time points t1 and t2 Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології, 2003, № 2 129 ing to few minutes of recording time were required. Like in our previous ap- proach a human expert could participate at step 3 were the number of spike classes were detected. 2.1 Spike detection from the noisy signal and derivative calculation methods. The derivatives of the signal were calculated following [1]. We repre- sent time signals used at Fig. 3–9. Fig. 3. 100 ms trial of brain activity observed in real experiment Fig. 4. 100 ms trial of first derivative with detected spikes Fig. 5. Spike templates one and its portraits in phase space. The centers of the spikes are at 0.41 ms A.I. Polyarush, A.S. Makarenko, I.V. Tetko ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 130 The estimation of the variance 2σ was calculated for the second derivative. The time intervals in which the latter exceeded σc were accepted as centers of potential spikes. Coefficient c was chosen by an expert manually with regard of spike amplitude and noise level. In our experiments, it equaled to 3. The derivatives of function ( )x t were estimated using integral operators ( ) ( )( )i i R D x t t f dα αω τ τ τ= −∫ , (11) where i αω is the i -th derivative of kernel function αω , which satisfies the fol- lowing conditions [1]: i. ,0)( =tαω if α>\|\| t ; Fig. 7. Spike templates three and its portraits in phase space. The centers of the spikes are at 0.41 ms Fig.. 8. Spike portraits in phase space for three modeled neurons. In other words, they are projections of vector fields on 2R , which describe the activity of each of the three neurons Fig. 6. Spike templates two and its portraits in phase space. The centers of the spikes are at 0.41 ms Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології, 2003, № 2 131 ii. ∫ = R dtt 1)(αω ; iii. αω has i continuous derivatives. The piece-wise polynomial kernel functions i αω were used (Fig. 9). The integral operator ( )txDi α acts also as a band-pass filter of the signal )(tx and its low and high cutoff frequencies are functions of parameter α. Since the duration of neuronal spikes are in the range of several ms, we selected pa- rameters α to have the low cutoff frequency of 1 kHz for both kernels. The same approach to select smoothing parameters was used in our previous study (1). 2.2. Distance function for phase trajectories used for estimation of spike classes. Let )(tx , where ( )bat ;∈ , be an arbitrary spike, )(1 tx and )(2 tx be the evaluation of its first and second derivatives respectively. Then ( ) ( )Ttxtxtx )(),( 21= describes the phase trajectory of that spike. In order to esti- mate the deviation of trajectory ( )ty from ( )tx we define such vector ( )tn , that ( ) ( ) ( )( )tytntx ω=+ for some ( )tω , and ( )tn is normal to ( )( )ty ω [1]. The norm of ( )tn can be calculated as ( )( ) [ ] [ ] ( ) ( ) ⎟ ⎠ ⎞⎜ ⎝ ⎛ ′−+′−=⋅⋅ ∩+−∈′ 2 22 211 ;; 2 )()()()(min)(),( tytxtytxtyxn bactctt . (12) Then we can introduce a difference function for phase trajectories of spikes )(tx and )(ty [1] as ( ) ( ) ( )( ))(),(~,)(),(~min)(),( ⋅⋅⋅⋅=⋅⋅ xydyxdyxd . (13) where ( ) ( ) ( )( ) ( )∫ ⋅⋅=⋅⋅ a dtttyxntwyxd 0 2)(),(min)(),(~ (14) and −α −3α /4 −α /2 −α /4 α /4 α /2 3α /4 α−α −α /2 α / 2 α ( )t1 αω ( )t2 αω Fig. 9. The kernel functions used to estimate the first and second derivative of the signal A.I. Polyarush, A.S. Makarenko, I.V. Tetko ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 132 ( ) [ ) [ ] ( ]⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − − ∈ = aa aa ta aa at a t tw ;, ,;,1 ,;0, 2 2 21 1 1 (15) is some weight function. In our implementation ms5,01 =a , ms12 =a and ms22 =a . For N detected spikes matrix of pair-wise distances between trajecto- ries { }jidD ,= , Nji ..1, = , where ( ) ( )( )⋅⋅= jiji xxdd ,, , was calculated. 2.3. Selection of the number of spike classes. On this step we formed classes of spikes using the set of spikes accumulated during few first minutes of recordings. Spikes generated by the same neuron must belong to one class. The difficult problem of this step is an absent of apriori information about the possi- ble number of spike classes (neurons) that are recorded by the microelectrode. An iterative procedure for detection the number of spike classes was used [1]. In the beginning each spike was considered as a new class. Let iC be the i- th class of spikes, js be the j -th accumulated spike, ic be the central spike of class iC , R be the radius of that class, i.e. the maximum allowed distance from the central spike to other spikes of iC . 1. A new center ci of class iC was found as ( )∑ ∈∈ = ikij Cs kj Cs i ssdc ,minarg . (16) 2. Spikes js with ( ) iii Rscd ≤, were added to the class, and spikes with ( ) Rscd ii >, were deleted from each class i . 3. Overlapped classes were determined. All pairs of classes iC , jC and their intersection ji CCI ∩= were determined. Each class jC was deleted if more than 50% of its spikes belonged to the class jC . 4. Steps 1, 2 and 3 were repeated. As it was shown in [1], such iterative process converges to stable centers of the classes. However, a larger training set is required for smaller values of R . 5. The remaining classes were analyzed by a human expert who performed further elimination of classes with small number of spikes. The user also selected the parameter R . Notice, that up to this step both our previous and new method were exactly the same. 2.4. Numerical representation of vector field. Vector fields were con- structed for each spike class. The vector field that characterized the activity of a neuron was described by phase trajectories of the most typical representatives of its class. The center of the class and several of the most close to it spikes were selected. The whole vector field was represented as a set of its integral curves. Let iV be a vector field that correspond to i -th neuron, and ( ) =tX ji, ( ) ( ) ( ) T ii i t dt xd t dt dx tx ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 2 ,, be a phase portrait of the j -th spike of the i -th class, Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології, 2003, № 2 133 i.e. one of the integral curves of the field iV . In order to obtain the value of vector ( )mVi for an arbitrary point Dm∈ all curves from the vector field are analyzed. A minimum Euclidean norm ( )0, tXm ji− is detected for some 0t and j and vector ( ) ( ) ( ) ( ) T iiiji t dt xd t dt xd t dt dx t dt Xd ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 3 3 2 2 , ,, is selected as an estimation of vec- tor ( )mVi . 2.5 Spike classification and probabilistic analysis of the deviation func- tion. Consider phase portrait ( ) Σ∈⋅ϕ of an arbitrary spike, vector field iV and some action of group G on it: ( )( ) ( )( )τϕϕ ττ −=⋅ tgtg )( . We propose the follow- ing iterative algorithm for calculation trajectory ( ) ( )( ) )( τϕϕ τ −⋅=′ tgt . (17) Consider 3 0 RS ∈ . Let ( )τσ −= tS0 . Let us choose step of integration t∆ . During the k -th iteration we calculate ( )kikk StVSS ∆+=+1 . (18) The iterations repeat until τ<∆tk . The last value of vector kS is the estimation of point ( )tσ of the transformed trajectory. Deviation function (9) was calculated for trajectory ( )tσ . Let us consider the distribution of (9). Spike ( )tx , which is a solution of system (1), can be repre- sented as ( ) ( ) ( )ttxtx ξ+= 0 , where ( )tx0 is some solution of (2) and ( )tξ is the influence of perturbation ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 2 2 ,,, dt xd dt dxxtF . Assume that ( )tξ is a random process with a zero mean and small correlation time. Taking into account that the signal was read at discrete time points spikes ( )tx , ( )tx0 and noise ( )tξ can be repre- sented as sequences jx , jx ,0 and jξ , where nj …1= . In our implementation we considered jξ as a random sequence of independent normally distributed values. The estimation of the first and second derivatives 1 jx and 2 jx are obtained as a linear combination (10) of jx : ( ) 11 ,0 1 ,0 1 ,0 111 jj k kk k kk k kkk k kkj xAxAxAxAx ξξξ +=+=+== ∑∑∑∑ , (19) ( ) 22 ,0 2 ,0 2 ,0 222 jj k kk k kk k kkk k kkj xAxAxAxAx ξξξ +=+=+== ∑∑∑∑ , (20) where 1 kA and 2 kA are the values at k -th time point of kernels 1 αω and 2 αω re- spectively, 1 ,0 jx and 2 ,0 jx are the estimation of the first and second derivatives of spike ( )tx0 , 1 jξ and 2 jξ are normally distributed random quantities with zero mean. A.I. Polyarush, A.S. Makarenko, I.V. Tetko ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 134 Consider transformed portrait (17) of spike ( )tx . Assume than ( )tx is close to ( )tx0 . Then during the iterative procedure (18) of transformation with vec- tor field the values of kS will stay close to phase portrait ( ) =tx0 ( ) ( ) ( ) T t dt xd t dt dx tx ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 0 2 0 0 ,, of spike ( )tx0 . Assuming that t∆ is the sampling interval and taking into consideration the numeric representation of vector field we can suppose that ( )Tjjjjjjji xxxxxxxV 2 ,0 2 1,0 1 ,0 1 1,0,01,0,0 ,,)( −−−≈ +++ . (21) The iteration (18) becomes ( )Tjjjjjjkk xxxxxxSS 2 ,0 2 1,0 1 ,0 1 1,0,01,01 ,, −−−+≈ ++++ . (22) Thus transformed with field iV portrait ( )tx′ of spike ( )tx is close to the tra- jectory described by jjx ξ ′+,0 , where ( )Tjjjj 21 ,, ξξξξ =′ . Then the deviation func- tion (9) squared for the first and second derivatives of spike ( )tx will equal to ( ) ( ) ( )∑ ⎟ ⎠ ⎞⎜ ⎝ ⎛ += k kkjx 22212 ξξατ . (23) It has 2χ distribution with known parameters. It makes possible to find a confidence interval δ with given confidence probability p for the spike class ob- served. If the value of ( )( )⋅σατ 2 does not exceed threshold δ then we consider spike ( )tx to be generated by the corresponding neuron, the activity of which is described by (1) and vector field iV . 3. NUMERICAL RESULTS ON SPIKES RECOGNITION Inverse Fourier transform was used to generate 50 seconds trial of an artificial noise with evenly distributed phases and magnitude that represented f/1 noise signature. The sampling frequency was 44 kHz. 3 different templates of spikes that were obtained in real experiment, in 10000 instances of every template, were imposed additively on the noise. Spikes did not overlap. The duration of every spike equaled to 2.43 ms, the duration of the whole experiment was 1097 s. Comparison of new and template matching method Symmetry group Template matching in phase space Template Unclassified Misclassified Error index Unclassified Misclassified Error index I 32 0 32 10 12 15.6 Ii 103 120 158.1 122 165 205.2 Iii 94 22 96.5 165 76 181.7 Spike separation based on symmetries analysis in phase space Системні дослідження та інформаційні технології, 2003, № 2 135 Unclassified — number of spikes of the given class that were not detected or were referred to another class. Misclassified – number of impulses that belong to another class or represent noise but were classified as spikes of the given class. ( ) ( )22 iedmisclassifedunclassifiErrorIndex += . 4. DISCUSSION The method described in the article is an elaboration of the method based on tem- plate matching in phase space [1]. The stages of spike detection and forming spike classes are generally similar to the latter. The principal difference of the method presented from template matching is the implementation of spike classifi- cation after spike classes are formed. The template matching method characterizes the whole class of spikes generated by the same neuron by the phase portrait of its typical spike. As contrasted to another methods [9], the method based on symme- tries analyses uses computational modeling of vector field that conforms the dif- ferential equation describing the activity of the chosen neuron and thus it is more stable against alterations of spike form. It showed better results than the former and can be useful in experiments, where spike tend to change their form. Acknowledgements. The research has been partially supported by INTAS Grant No.97-168. REFERENCE 1. Aksenova T.I., Chibirova O.K., Dryga O.A., Tetko T.V., Benabid A–C., Villa A. E. An unsupervised automatic method for sorting neuronal spike waveform in awake and freely moving animals // Methods. — 30. — 2003. — P. 178–187. 2. Warner F.W. Foundations of Differentiable Manifolds and Lie Groups. Springer- Verlag, 1983. — 267 p. 3. Letelier J.C., Weber P.P., Spike Sorting Based on Discrete Wavelet Transform Coef- ficients // Journal of Neuroscience Methods, 101. — 2000. — P. 93–106. 4. Aksenova, T. I.; Shelekhova, V. Yu., Fast Derivative Estimation Algorithm Using In- tegral Convolution Operators // Cybernetics and Computing Technology. — 107. 1997. — P. 62–71 5. Aksenova, T. I., Shelekhova, V. Yu. The Use of Integral Convolution Operator for Approximation of Derivatives by Means of Functions of Preset Smoothness // Cybernetics and Computing Technology. — 103. — 1994. — P. 79–82. 6. Gudzenko L. I. Statistical method for self-oscillating system characteristics detection // Izvestiia Vuzov Radiophysics. — 1962. — 5. — P. 573–587 (In Russian). 7. Makarenko A. Complexity of individual objects without Including Probability Con- cept: Asymmetry, Observer, and Perception Problems // AIP Conference Pro- ceedings. — 2001. — 573. — P. 201–215. 8. Makarenko A.S., Polyarush A.I., Krachkovsky V. Pattern recognition and classifica- tion investigations of neural system cells. Abstracts of I Nation // Conf. of stu- dents and young researchers «System Analysis and Informational Technologies», Kiev, 28–29, June 1999. — SNK IASA. — Kiev, 1999. — P. 42–43. 9. Rosenblatt F. Principles of neurodynamics, perceptrons and the theory of brain me- chanics. — Washington D.C.: Spartan Books, 1962. — 560 p. Received 24.01.2003

Spike separation based on symmetries analysis in phase space

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