Skip to main content
Log in

Towards dynamical system models of language-related brain potentials

  • Research Article
  • Published:
Cognitive Neurodynamics Aims and scope Submit manuscript

Abstract

Event-related brain potentials (ERP) are important neural correlates of cognitive processes. In the domain of language processing, the N400 and P600 reflect lexical-semantic integration and syntactic processing problems, respectively. We suggest an interpretation of these markers in terms of dynamical system theory and present two nonlinear dynamical models for syntactic computations where different processing strategies correspond to functionally different regions in the system’s phase space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. Ungrammatical sentences are usually denoted by the * in linguistic examples.

  2. Also an early left anterior negativity (ELAN) was observed for such violations (Friederici et al. 1993).

  3. Note that we modified Smolensky’s original definition slightly: The ordered pairs (f, r) are the definiens of the filler/role binding f/r(s), the definiendum, which therefore has to stand in front of the set definition symbol “|”.

  4. We shall see below, that the sum in Eq. 27 has to be replaced by the direct sum over tensor product spaces for a proper treatment of recursion.

  5. Words u = u 1,…, u p , v = v 1,…, v q of finite length formed from symbols in T, can be concatenated to form a new word \(u \cdot v = u_{1}, \ldots, u_{p} v_{1}, \ldots, v_{q}\in{\mathbf{T}}^{\ast}.\) This concatenation product is associative: u ·(v ·w) = (u ·v) ·w, such that (T*,·) forms a semigroup. Note that the concatenation is generally not commutative, thus justifying the notion of a “quantum operator”.

References

  • Aho AV, Ullman JD (1972) The theory of parsing, translation and compiling, vol i: parsing. Prentice Hall, Englewood Cliffs (NJ)

    Google Scholar 

  • Allefeld C, Frisch S, Schlesewsky M (2004) Detection of early cognitive processing by event-related phase synchronization analysis. NeuroReport 16(1):13–16

    Article  Google Scholar 

  • Amari SI (1974) A method of statistical neurodynamics. Kybernetik 14:201–215

    PubMed  CAS  Google Scholar 

  • Amari SI (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybernet 27:77–87

    Article  CAS  Google Scholar 

  • Anderson JR, Bothell D, Byrne MD, Douglass S, Lebiere C, Qin Y (2004) An integrated theory of the mind. Psychol Rev 111(4):1036–1060

    Article  PubMed  Google Scholar 

  • Atmanspacher H, beim Graben P (2007) Contextual emergence of mental states from neurodynamics. Chaos Complex Lett 2(2/3):151–168

    Google Scholar 

  • Başar E (1980) EEG-brain dynamics. Relations between EEG and brain evoked potentials. Elsevier/North Holland Biomedical Press, Amsterdam

    Google Scholar 

  • Başar E (1998) Brain function and oscillations. Vol I: brain oscillations. Principles and approaches. Springer series in synergetics. Springer, Berlin

    Google Scholar 

  • Bader M, Meng M (1999) Subject–object ambiguities in German embedded clauses: an across-the-board comparison. J Psycholinguist Res 28(2):121–143

    Article  Google Scholar 

  • Beer RD (2000) Dynamical approaches to cognitive science. Trends Cogn Sci 4(3):91–99

    Article  PubMed  Google Scholar 

  • Bornkessel I, Schlesewsky M (2006) The extended argument dependency model: a neurocognitive approach to sentence comprehension across languages. Psychol Rev 113(4):787–821

    Article  PubMed  Google Scholar 

  • Bornkessel I, McElree B, Schlesewsky M, Friederici AD (2004) Multi-dimensional contributions to garden path strength: dissociating phrase structure from case marking. J Mem Lang 51:494–522

    Article  Google Scholar 

  • Boston MF, Hale JT, Kliegl R, Patil U, Vasishth S (in press) Parsing costs as predictors of reading difficulty: an evaluation using the Potsdam Sentence Corpus. J Eye Mov Res 1

  • Chomsky N (1981) Lectures on goverment and binding. Foris

  • Christiansen MH, Chater N (1999) Connectionist natural language processing: the state of the art. Cogn Sci 23(4):417–437

    Article  Google Scholar 

  • Coles MGH, Rugg MD (1995) Event-related brain potentials: an introduction. In: Coles MGH, Rugg MD (eds) Electrophysiology of mind: event-related brain potentials and cognition, chap 1. Oxford University Press, Oxford

  • Crutchfield JP (1994) The calculi of emergence: computation, dynamics and induction. Physica D 75:11–54

    Article  Google Scholar 

  • Cvitanović P, Gunaratne GH, Procaccia I (1988) Topological and metric properties of Hénon-type strange attractors. Phys Rev A 38(3):1503–1520

    Article  PubMed  Google Scholar 

  • Dambacher M, Kliegl R, Hofmann M, Jacobs AM (2006) Frequency and predictability effects on event-related potentials during reading. Brain Res 1084:89–103

    Article  PubMed  CAS  Google Scholar 

  • Dolan CP, Smolensky P (1989) Tensor product production system: a modular architecture and representation. Connect Sci 1(1):53–68

    Article  Google Scholar 

  • Drenhaus H, beim Graben P, Saddy D, Frisch S (2006) Diagnosis and repair of negative polarity constructions in the light of symbolic resonance analysis. Brain Lang 96(3):255–268

    Article  PubMed  Google Scholar 

  • Elman JL (1995) Language as a dynamical system. In: Port, van Gelder (eds), pp 195–223

  • Erlhagen W, Schöner G (2002) Dynamic field theory of movement preparation. Psychol Rev 109(3):545–572

    Article  PubMed  Google Scholar 

  • Fodor JD, Ferreira F (eds) (1998) Reanalysis in sentence processing. Kluwer, Dordrecht

    Google Scholar 

  • Fodor JD, Frazier L (1980) Is the human sentence parsing mechanism an ATN? Cognition 6:417–459

    Article  Google Scholar 

  • Fodor J, Pylyshyn ZW (1988) Connectionism and cognitive architecture: a critical analysis. Cognition 28:3–71

    Article  PubMed  CAS  Google Scholar 

  • Frazier L, Fodor JD (1978) The sausage machine: a new two-stage parsing model. Cognition 6:291–326

    Article  Google Scholar 

  • Freeman WJ (2007) Definitions of state variables and state space for brain-computer interface. Part 1. Multiple hierarchical levels of brain function. Cogn Neurodyn 1:3–14

    Article  PubMed  Google Scholar 

  • Friederici AD (1995) The time course of syntactic activation during language processing: a model based on neuropsychological and neurophysiological data. Brain Lang 50:259–281

    Article  PubMed  CAS  Google Scholar 

  • Friederici AD (1998) Diagnosis and reanalysis: two processing aspects the brain may differentiate. In: Fodor, Ferreira (eds), pp 177–200

  • Friederici AD (1999) The neurobiology of language comprehension. In: Friederici AD (ed) Language comprehension: a biological perspective, 2nd edn. Springer, Berlin, pp 265–304

    Google Scholar 

  • Friederici AD (2002) Towards a neural basis of auditory language processing. Trends Cogn Sci 6:78–84

    Article  PubMed  Google Scholar 

  • Friederici AD, Pfeifer E, Hahne (1993) Event-related brain potentials during natural speech processing: effects of semantic morphological and syntactic violations. Cogn Brain Res 1:183–192

  • Friederici AD, Steinhauer K, Mecklinger A, Meyer M (1998) Working memory constraints on syntactic ambiguity resolution as revealed by electrical brain responses. Biol Psychol 47:193–221

    Article  PubMed  CAS  Google Scholar 

  • Friederici AD, Mecklinger A, Spencer KM, Steinhauer K, Donchin E (2001) Syntactic parsing preferences and their on-line revisions: a spatio-temporal analysis of event-related brain potentials. Cogn Brain Res 11:305–323

    Article  CAS  Google Scholar 

  • Frisch S, beim Graben P (2005) Finding needles in haystacks: symbolic resonance analysis of event-related potentials unveils different processing demands. Cogn Brain Res 24(3):476–491

    Article  Google Scholar 

  • Frisch S, Schlesewsky M (2001) The N400 reflects problems of thematic hierarchizing. NeuroReport 12(15):3391–3394

    Article  PubMed  CAS  Google Scholar 

  • Frisch S, Schlesewsky M, Saddy D, Alpermann A (2002) The P600 as an indicator of syntactic ambiguity. Cognition 85:B83–B92

    Article  PubMed  Google Scholar 

  • Frisch S, beim Graben P, Schlesewsky M (2004) Parallelizing grammatical functions: P600 and P345 reflect different cost of reanalysis. Int J Bifurcat Chaos 14(2):531–549

    Article  Google Scholar 

  • Frisch S, Kotz SA, Friederici AD (2008) Neural correlates of normal and pathological language processing. In: Ball MJ, Perkins M, Müller N, Howard S (eds) Handbook of clinical linguistics. Blackwell, Boston

  • Garagnani M, Wennekers T, Pulvermüller F (2007) A neuronal model of the language cortex. Neurocomputing 70:1914–1919

    Article  Google Scholar 

  • van Gelder T (1998) The dynamical hypothesis in cognitive science. Behav Brain Sci 21(5):615–628

    PubMed  Google Scholar 

  • Gerth S (2006) Parsing mit minimalistischen, gewichteten Grammatiken und deren Zustandsraumdarstellung. Master’s thesis, Universität Potsdam

  • beim Graben P (2001) Estimating and improving the signal-to-noise ratio of time series by symbolic dynamics. Phys Rev E 64:051104

    Article  CAS  Google Scholar 

  • beim Graben P (2004) Incompatible implementations of physical symbol systems. Mind Matter 2(2):29–51

    Google Scholar 

  • beim Graben P (2006) Pragmatic information in dynamic semantics. Mind Matter 4(2):169–193

    Google Scholar 

  • beim Graben P, Frisch S (2004) Is it positive or negative? On determining ERP components. IEEE Trans Biomed Eng 51(8):1374–1382

    Article  Google Scholar 

  • beim Graben P, Saddy D, Schlesewsky M, Kurths J (2000) Symbolic dynamics of event-related brain potentials. Phys Rev E 62(4):5518–5541

    Article  CAS  Google Scholar 

  • beim Graben P, Jurish B, Saddy D, Frisch S (2004) Language processing by dynamical systems. Int J Bifurcat Chaos 14(2):599–621

    Article  Google Scholar 

  • beim Graben P, Frisch S, Fink A, Saddy D, Kurths J (2005) Topographic voltage and coherence mapping of brain potentials by means of the symbolic resonance analysis. Phys Rev E 72:051916

    Article  CAS  Google Scholar 

  • beim Graben P, Gerth S, Saddy D, Potthast R (2007) Fock space representations in neural field theories. In: Biggs N, Bonnet-Bendhia AS, Chamberlain P, Chandler-Wilde S, Cohen G, Haddar H, Joly P, Langdon S, Lunéville E, Pelloni B, Potherat D, Potthast R (eds) Proc. waves 2007. The 8th international conference on mathematical and numerical aspects of waves. Dept. of Mathematics, University of Reading, Reading, pp 120–122

  • Grodzinsky Y, Friederici AD (2006) Neuroimaging of syntax and syntactic processing. Curr Opin Neurobiol 16:240–246

    Article  PubMed  CAS  Google Scholar 

  • Haag R (1992) Local quantum physics: fields, particles, algebras. Springer, Berlin

    Google Scholar 

  • Haegeman L (1994) Introduction to goverment & binding theory, Blackwell textbooks in linguistics, vol 1, 2nd edn. Blackwell Publishers, Oxford, 1st edition 1991

  • Hagoort P (2003) How the brain solves the binding problem for language: a neurocomputational model of syntactic processing. NeuroImage 20:S18–S29

    Article  PubMed  Google Scholar 

  • Hagoort P (2005) On Broca, brain, and binding: A new framework. Trends Cogn Sci 9(9):416–423

    PubMed  Google Scholar 

  • Hagoort P, Brown CM, Groothusen J (1993) The syntactic positive shift (SPS) as an ERP measure of syntactic processing. Lang Cogn Process 8:439–483

    Article  Google Scholar 

  • Hale JT (2003) The information conveyed by words in sentences. J Psycholinguist Res 32(2):101–123

    Article  PubMed  Google Scholar 

  • Hale JT (2006) Uncertainty about the rest of the sentence. Cogn Sci 30(4)

  • Hale JT, Smolensky P (2006) Harmonic grammar and harmonic parsers for formal languages. In: Smolensky, Legendre (eds), chap 10, pp 393–415

  • Hao BL (1989) Elementary symbolic dynamics and chaos in dissipative systems. World Scientific, Singapore

    Google Scholar 

  • Hopcroft JE, Ullman JD (1979) Introduction to automata theory, languages, and computation. Addison–Wesley, Menlo Park, California

    Google Scholar 

  • Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, Yen NC, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc Lond A 454:903–995

    Article  Google Scholar 

  • Jirsa VK, Haken H (1996) Field theory of electromagnetic brain activity. Phys Rev Lett 77(5):960–963

    Article  PubMed  CAS  Google Scholar 

  • Kaan E, Harris A, Gibson E, Holcomb P (2000) The P600 as an index of syntactic integration difficulty. Lang Cogn Process 15(2):159–201

    Article  Google Scholar 

  • Kandel ER, Schwartz JH, Jessel TM (eds) (1995) Essentials of neural science and behavior. Appleton & Lange, East Norwalk, Connecticut

  • Kennel MB, Buhl M (2003) Estimating good discrete partitions from observed data: Symbolic false nearest neighbors. Phys Rev Lett 91(8):084–102

    Article  CAS  Google Scholar 

  • Kutas M, Hillyard SA (1980) Reading senseless sentences: brain potentials reflect semantic incongruity. Science 207:203–205

    Article  PubMed  CAS  Google Scholar 

  • Kutas M, Hillyard SA (1984) Brain potentials during reading reflect word expectancy and semantic association. Nature 307:161–163

    Article  PubMed  CAS  Google Scholar 

  • Kutas M, van Petten CK (1994) Psycholinguistics electrified. Event-related brain potential investigations. In: Gernsbacher MA (ed) Handbook of psycholinguistics. Academic Press, San Diego, pp 83–133

    Google Scholar 

  • Lewis RL (1998) Reanalysis and limited repair parsing: leaping off the garden path. In: Fodor, Ferreira (eds), pp 247–285

  • Lewis RL (2000) Computational psycholinguistics. In: Encyclopedia of cognitive science, Macmillan Reference Ltd

  • Lewis RL, Vasishth S (2006) An activation-based model of sentence processing as skilled memory retrieval. Cogn Sci 29:375–419

    Google Scholar 

  • Lewis RL, Vasishth S, Van Dyke J (2006) Computational principles of working memory in sentence comprehension. Trends Cogn Sci 10:447–454

    Article  PubMed  Google Scholar 

  • Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (UK), reprint 1999

  • Makeig S, Westerfield M, Jung TP, Enghoff S, Townsend J, Courchesne E, Sejnowski TJ (2002) Dynamic brain sources of visual evoked responses. Science 295:690–694

    Article  PubMed  CAS  Google Scholar 

  • Marcus M (1980) A theory of syntactic recognition for natural language. MIT Press, Cambrigde (MA)

    Google Scholar 

  • Marwan N, Meinke A (2004) Extended recurrence plot analysis and its application to ERP data. Int J Bifurcat Chaos 14(2):761–771

    Article  Google Scholar 

  • McElree B (2000) Sentence comprehension is mediated by content-addressable memory structures. J Psycholinguist Res 29(2):111–123

    Article  PubMed  CAS  Google Scholar 

  • McElree B, Dosher BA (1993) Serial retrieval processes in the recovery of order information. J Exp Psychol Gen 122(3):291–315

    Article  Google Scholar 

  • Mecklinger A, Schriefers H, Steinhauer K, Friederici AD (1995) Processing relative clauses varying on syntactic and semantic dimensions: an analysis with event-related potentials. J Mem Lang 23:477–494

    CAS  Google Scholar 

  • Michaelis J (2001) Derivational minimalism is mildly context-sensitive. In: Moortgat M (ed) Logical aspects of computational linguistics. Lecture notes in artificial intelligence, vol 2014, Springer, Berlin, pp 179–198

  • Mizraji E (1989) Context-dependent associations in linear distributed memories. Bull Math Biol 51(2):195–205

    PubMed  CAS  Google Scholar 

  • Mizraji E (1992) Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets Syst 50:179–185

    Article  Google Scholar 

  • Moore C (1990) Unpredictability and undecidability in dynamical systems. Phys Rev Lett 64(20):2354–2357

    Article  PubMed  Google Scholar 

  • Moore C (1991) Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4:199–230

    Article  Google Scholar 

  • Moore C (1998) Dynamical recognizers: Real-time language recognition by analog computers. Theor Comput Sci 201:99–136

    Article  Google Scholar 

  • Moore C, Crutchfield JP (2000) Quantum automata and quantum grammars. Theor Comput Sci 237:275–306

    Article  Google Scholar 

  • Neville HJ, Nicol J, Barss A, Forster K, Garrett M (1991) Syntactically based sentence processing classes: Evidence from event-related potentials. J Cogn Neurosci 6:233–244

    Google Scholar 

  • Newell A, Simon HA (1976) Computer science as empirical inquiry: symbols and search. Commun Assoc Comput Mach 19:113–126

    Google Scholar 

  • Niedermeyer E, da Silva FHL (eds) (1999) Electroencephalography. Basic principles, clinical applications, and related fields, 4th edn. Lippincott Williams and Wilkins, Baltimore

  • Osterhout L, Holcomb PJ (1992) Event-related brain potentials elicited by syntactic anomaly. J Mem Lang 31:785–806

    Article  Google Scholar 

  • Osterhout L, Holcomb PJ (1995) Event-related potentials and language comprehension. In: Coles MGH, Rugg MD (eds) Electrophysiology of mind: event-related brain potentials and cognition, chap 6. Oxford University Press, Oxford

  • Osterhout L, Holcomb PJ, Swinney DA (1994) Brain potentials elicited by garden-path sentences: evidence of the application of verb information during parsing. J Exp Psychol Learn Mem Cogn 20(4):786–803

    Article  PubMed  CAS  Google Scholar 

  • Pollack JB (1991) The induction of dynamical recognizers. Mach Learn 7:227–252. Also published in Port and van Gelder (1995), pp 283–312.

    Google Scholar 

  • Port RF, van Gelder T (eds) (1995) Mind as motion: explorations in the dynamics of cognition. MIT Press, Cambridge (MA)

    Google Scholar 

  • Regan D (1989) Human brain electrophysiology: evoked potentials and evoked magnetic fields in science and medicine. Elsevier, New York

    Google Scholar 

  • Rumelhart DE, McClelland JL, the PDP Research Group (eds) (1986) Parallel distributed processing: explorations in the microstructure of cognition, vol I. MIT Press, Cambridge (MA)

  • Schinkel S, Marwan N, Kurths J (2007) Order patterns recurrence plots in the analysis of ERP data. Cogn Neurodyn. doi:10.1007/s11571-007-9023-z

  • Schlesewsky M, Bornkessel I (2006) Context-sensitive neural responses to conflict resolution: electrophysiological evidence from subject–object ambiguities in language comprehension. Brain Res 1098:139–152

    Article  PubMed  CAS  Google Scholar 

  • Shannon CE, Weaver W (1949) The mathematical theory of communication. University of Illinois Press, Urbana, reprint 1963

  • Sharbrough F, Chartrian GE, Lesser RP, Lüders H, Nuwer M, Picton TW (1995) American Electroencephalographic Society guidelines for standard electrode position nomenclature. J Clin Neurophysiol 8:200–202

    Google Scholar 

  • Shieber SM (1985) Evidence against the context-freeness of natural language. Linguist Philos 8:333–343

    Article  Google Scholar 

  • Siegelmann HT (1996) The simple dynamics of super Turing theories. Theor Comput Sci 168:461–472

    Article  Google Scholar 

  • Smolensky P (1990) Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artif Intell 46:159–216

    Article  Google Scholar 

  • Smolensky P (1991) Connectionism, constituency, and the language of thought. In: Loewer B, Rey G (eds) Meaning in mind. Fodor and his critics, chap 12. Blackwell, Oxford, pp 201–227

  • Smolensky P (2006) Harmony in linguistic cognition. Cogn Sci 30:779–801

    Article  Google Scholar 

  • Smolensky P, Legendre G (2006) The harmonic mind. From neural computation to optimality-theoretic grammar, vol 1: cognitive architecture. MIT Press, Cambridge (MA)

    Google Scholar 

  • Stabler EP (1997) Derivational minimalism. In: Retoré C (eds) Logical aspects of comutational linguistics, Springer lecture notes in computer science, vol 1328. Springer, New York, pp 68–95

    Chapter  Google Scholar 

  • Stabler EP, Keenan EL (2003) Structural similarity within and among languages. Theor Comput Sci 293:345–363

    Article  Google Scholar 

  • Staudacher P (1990) Ansätze und Probleme prinzipienorientierten Parsens. In: Felix SW, Kanngießer S, Rickheit G (eds) Sprache und Wissen. Westdeutscher Verlag, Opladen, pp 151–189

    Google Scholar 

  • Sweeney-Reed CM, Nasuto SJ (2007) A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition. J Cogn Neurosci. doi:10.1007/s10827-007-0020-3

  • Tabor W (1998) Dynamical automata. Technical report TR98-1694, Cornell Computer Science Department, Department of Psychology, Uris Hall, Cornell University, Ithaca, NY 14853

  • Tabor W (2000) Fractal encoding of context-free grammars in connectionist networks. Expert Syst Int J Knowl Eng Neural Networ 17(1):41–56

    Google Scholar 

  • Tabor W, Tanenhaus MK (1999) Dynamical models of sentence processing. Cogn Sci 23(4):491–515

    Article  Google Scholar 

  • Tabor W, Juliano C, Tanenhaus MK (1997) Parsing in a dynamical system: an attractor-based account of the interaction of lexical and structural constraints in sentence processing. Lang Cogn Process 12(2/3):211–271

    Article  Google Scholar 

  • Thelen E, Schöner G, Scheier C, Smith LB (2001) The dynamics of embodiment: a field theory of infant perseverative reaching. Behav Brain Sci 24:1–86

    Article  PubMed  CAS  Google Scholar 

  • van Valin R (1993) A synopsis of role and reference grammar. In: van Valin R (eds) Advances in role and reference grammar. Benjamins, Amsterdam

    Google Scholar 

  • Vasishth S, Lewis RL (2006a) Argument-head distance and processing complexity: explaining both locality and antilocality effects. Language 82

  • Vasishth S, Lewis RL (2006b) Human language processing: symbolic models. In: Brown K (eds) Encyclopedia of language and linguistics, vol 5. Elsevier, Amsterdam, pp 410–419

    Google Scholar 

  • Vasishth S, Brüssow S, Lewis RL, Drenhaus H (2008) Processing polarity: how the ungrammatical intrudes on the grammatical. Cogn Sci 32(4)

  • van der Velde F, de Kamps M (2006) Neural blackboard architectures of combinatorial structures in cognition. Behav Brain Sci 29:37–108

    PubMed  Google Scholar 

  • Vos SH, Gunter TC, Schriefers H, Friederici AD (2001) Syntactic parsing and working memory: the effects of syntactic complexity, reading span, and concurrent load. Lang Cogn Process 16(1):65–103

    Article  Google Scholar 

  • Vosse T, Kempen G (2000) Syntactic structure assembly in human parsing: a computational model based on competitive inhibition and a lexicalist grammar. Cognition 75:105–143

    Article  PubMed  CAS  Google Scholar 

  • van der Waerden BL (2003) Algebra, vol 2. Springer, New York

    Google Scholar 

  • Wegner P (1998) Interactive foundations of computing. Theor Comput Sci 192:315–351

    Article  Google Scholar 

  • Wennekers T, Garagnani M, Pulvermüller F (2006) Language models based on hebbian cell assemblies. J Physiol (Paris) 100:16–30

    Article  Google Scholar 

  • Wright JJ, Rennie CJ, Lees GJ, Robinson PA, Bourke PD, Chapman CL, Gordon E, Rowe DL (2004) Simulated electrocortical activity at microscopic, mesoscopic and global scales. Int J Bifurcat Chaos 14(2):853–872

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We thank Stefan Frisch for support conducting the ERP experiment and for helpful comments on the manuscript. We also thank Leticia Pablos Robles, Roland Potthast, and Slawomir Nasuto for inspiring discussions. The ERP pilot study was funded through grant FOR 375/1-4 by Deutsche Forschungsgemeinschaft.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Peter beim Graben.

Appendices

Appendix

Trajectories of the tensor product top-down recognizer

The processing trajectory for the \({\tt so}\) sentence (33) according to grammar G 1 (\(\tau_1({\tt so})\)) comes out to be

$$ \begin{aligned} &\user2{f}_8 \otimes \user2{r}_1 \otimes \user2{s}_1 \oplus\user2{f}_8 \otimes \user2{s}_2 \oplus [\user2{f}_1 \oplus \user2{f}_2 \oplus \user2{f}_3\oplus \user2{f}_4 \oplus \user2{f}_3 \oplus \user2{f}_6 \oplus \user2{f}_7 ] \otimes\user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_1 \otimes\user2{s}_4\mathop{\to}\limits^{(35)}\\ &\user2{f}_8 \otimes \user2{r}_1 \oplus\user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_3 \otimes\user2{s}_1 \oplus \user2{f}_9 \otimes \user2{s}_2 \oplus [\user2{f}_2 \oplus \user2{f}_3\oplus \user2{f}_4 \oplus \user2{f}_3 \oplus \user2{f}_6 \oplus \user2{f}_7 ] \otimes\user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_2 \otimes{\mathbf{s}}_4\mathop{\to}\limits^{(36)}\\ &\user2{f}_8 \otimes \user2{r}_1 \oplus\user2{f}_1 \otimes \user2{r}_2 \oplus [\user2{f}_9 \otimes \user2{r}_1 \oplus\user2{f}_2 \otimes \user2{r}_2 \oplus \user2{f}_{13} \otimes \user2{r}_3] \otimes\user2{r}_3 \otimes \user2{s}_1 \oplus \user2{f}_{13} \otimes \user2{s}_2 \oplus[\user2{f}_2 \oplus \user2{f}_3 \oplus \user2{f}_4 \oplus \user2{f}_3 \oplus \user2{f}_6\oplus \user2{f}_7 ] \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_2 \otimes\user2{s}_4 \to \\ & \user2{f}_8 \otimes \user2{r}_1 \oplus \user2{f}_1 \otimes\user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1 \otimes \user2{r}_3 \oplus\user2{f}_2\otimes \user2{r}_2 \otimes \user2{r}_3 \oplus \user2{f}_{13} \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{s}_1 \oplus \user2{f}_{13} \otimes \user2{s}_2 \oplus[\user2{f}_3 \oplus\user2{f}_4 \oplus \user2{f}_3 \oplus \user2{f}_6 \oplus \user2{f}_7 ]\otimes \user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_3 \otimes\user2{s}_4\mathop{\to}\limits^{(37)} \\ & \user2{f}_8 \otimes \user2{r}_1 \oplus \user2{f}_1\otimes \user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1 \otimes \user2{r}_3\oplus\user2{f}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \oplus [\user2{f}_{13} \otimes\user2{r}_1 \oplus \user2{f}_{3} \otimes \user2{r}_2 \oplus \user2{f}_{10} \otimes\user2{r}_3 ] \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{s}_1 \oplus\user2{f}_{10} \otimes \user2{s}_2 \oplus \\ & [\user2{f}_3 \oplus \user2{f}_4 \oplus\user2{f}_3 \oplus \user2{f}_6 \oplus \user2{f}_7 ] \otimes \user2{r}_1 \otimes\user2{s}_3 \oplus \user2{f}_3 \otimes \user2{s}_4 \to \\ & \user2{f}_8 \otimes\user2{r}_1 \oplus \user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1\otimes \user2{r}_3 \oplus \user2{f}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \oplus\user2{f}_{13} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{3} \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{10} \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{s}_1 \oplus \user2{f}_{10} \otimes \user2{s}_2 \oplus \\ & [\user2{f}_4 \oplus\user2{f}_3 \oplus \user2{f}_6 \oplus \user2{f}_7 ] \otimes \user2{r}_1 \otimes\user2{s}_3 \oplus \user2{f}_4 \otimes \user2{s}_4 \mathop{\to}\limits^{(38)} \\ &\user2{f}_8 \otimes \user2{r}_1 \oplus \user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9\otimes \user2{r}_1 \otimes \user2{r}_3 \oplus \user2{f}_2 \otimes \user2{r}_2 \otimes\user2{r}_3 \oplus \user2{f}_{13} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{3} \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \\ & [\user2{f}_{10} \otimes \user2{r}_1 \oplus \user2{f}_{12} \otimes\user2{r}_2 \oplus \user2{f}_{7} \otimes \user2{r}_3 ] \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{s}_1 \oplus \user2{f}_{12} \otimes\user2{s}_2 \oplus [\user2{f}_4 \oplus \user2{f}_3 \oplus \user2{f}_6 \oplus \user2{f}_7] \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_4 \otimes \user2{s}_4 \to \\ &\user2{f}_8 \otimes \user2{r}_1 \oplus \user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9\otimes \user2{r}_1 \otimes \user2{r}_3 \oplus \user2{f}_2 \otimes \user2{r}_2 \otimes\user2{r}_3 \oplus \user2{f}_{13} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{3} \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{10} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \\ & \user2{f}_{12} \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{7} \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{s}_1 \oplus \user2{f}_{12} \otimes \user2{s}_2 \oplus [\user2{f}_4 \oplus\user2{f}_3 \oplus \user2{f}_6 ] \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus\user2{f}_4 \otimes \user2{s}_4 \mathop{\to}\limits^{(39)}\\ & \user2{f}_8 \otimes\user2{r}_1 \oplus \user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1\otimes \user2{r}_3 \oplus \user2{f}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \oplus\user2{f}_{13} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{3} \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{10} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \\ & [\user2{f}_{12} \otimes \user2{r}_1 \oplus \user2{f}_{4} \otimes\user2{r}_2 \oplus \user2{f}_{14} \otimes \user2{r}_3 ] \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{7} \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{s}_1 \oplus \user2{f}_{14} \otimes \user2{s}_2 \oplus [\user2{f}_4 \oplus\user2{f}_3 \oplus \user2{f}_6 ] \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus\user2{f}_4 \otimes \user2{s}_4 \to \\ & \user2{f}_8 \otimes \user2{r}_1 \oplus\user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1 \otimes\user2{r}_3 \oplus \user2{f}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \oplus\user2{f}_{13} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{3} \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{10} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \\ & \user2{f}_{12} \otimes \user2{r}_1 \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{4} \otimes\user2{r}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{14} \otimes \user2{r}_3 \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{7} \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{s}_1 \\ & \oplus \user2{f}_{14} \otimes \user2{s}_2 \oplus [\user2{f}_3 \oplus\user2{f}_6 ] \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_3 \otimes\user2{s}_4 \mathop{\to}\limits^{(40)}\\ & \user2{f}_8 \otimes \user2{r}_1 \oplus\user2{f}_1 \otimes \user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1 \otimes\user2{r}_3 \oplus \user2{f}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \oplus\user2{f}_{13} \otimes \user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{3} \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \\ &\user2{f}_{10} \otimes\user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{12} \otimes \user2{r}_1 \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{4} \otimes\user2{r}_2 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \\ &[\user2{f}_{14} \otimes \user2{r}_1 \oplus \user2{f}_{3} \otimes\user2{r}_2 \oplus \user2{f}_{6} \otimes \user2{r}_3 ] \otimes \user2{r}_3 \otimes\user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{7} \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{s}_1 \oplus 0 \otimes \user2{s}_2 \oplus [\user2{f}_3 \oplus\user2{f}_6 ] \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_3 \otimes\user2{s}_4 \to \\ & \user2{f}_8 \otimes \user2{r}_1 \oplus \user2{f}_1 \otimes\user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1 \otimes \user2{r}_3 \oplus \user2{f}_2\otimes \user2{r}_2 \otimes \user2{r}_3 \oplus \user2{f}_{13} \otimes \user2{r}_1 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{3} \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \\ & \user2{f}_{10} \otimes \user2{r}_1 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{12} \otimes\user2{r}_1 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{4} \otimes \user2{r}_2 \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \\ &[\user2{f}_{14} \otimes\user2{r}_1 \oplus \user2{f}_{3} \otimes \user2{r}_2 \oplus \user2{f}_{6} \otimes\user2{r}_3 ] \otimes \user2{r}_3 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{7} \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{s}_1 \oplus 0 \otimes\user2{s}_2 \oplus \user2{f}_6 \otimes \user2{r}_1 \otimes \user2{s}_3 \oplus \user2{f}_6\otimes \user2{s}_4 \to \\ & \user2{f}_8 \otimes \user2{r}_1 \oplus \user2{f}_1 \otimes\user2{r}_2 \oplus \user2{f}_9 \otimes \user2{r}_1 \otimes \user2{r}_3 \oplus \user2{f}_2\otimes \user2{r}_2 \otimes \user2{r}_3 \oplus \user2{f}_{13} \otimes \user2{r}_1 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{3} \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{10} \otimes \user2{r}_1 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \\ & \user2{f}_{12} \otimes\user2{r}_1 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{4} \otimes \user2{r}_2 \otimes \user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus \user2{f}_{14} \otimes\user2{r}_1 \otimes \user2{r}_3 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{r}_3 \oplus \\ & \user2{f}_{3} \otimes\user2{r}_2 \otimes\user2{r}_3 \otimes \user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \oplus \user2{f}_{6} \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_2 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \oplus\user2{f}_{7} \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes \user2{r}_3 \otimes\user2{r}_3 \otimes \user2{s}_1 \oplus 0 \otimes \user2{s}_2 \oplus 0 \otimes \user2{s}_3\oplus 0 \otimes \user2{s}_4 \\ \end{aligned} $$

On the other hand, processing the \({\tt os}\) sentence (34) with grammar \(G_1(\tau_1({\tt os}))\) leads into the garden path. Its trajectory is then

$$ \begin{aligned} & \user2{f}_{8} \otimes \user2{r}_{1} \otimes \user2{s}_1 \oplus \user2{f}_{8} \otimes \user2{s}_2 \oplus [\user2{f}_{1} \oplus \user2{f}_{2} \oplus \user2{f}_{5} \oplus \user2{f}_{3} \oplus \user2{f}_{3} \oplus \user2{f}_{6} \oplus \user2{f}_{7}] \otimes \user2{r}_{1} \otimes \user2{s}_3 \oplus \user2{f}_1 \otimes \user2{s}_4 \mathop{\to}\limits^{(35)} \\ & [\user2{f}_{8} \otimes \user2{r}_{1} \oplus \user2{f}_{1} \otimes \user2{r}_{2} \oplus \user2{f}_{9} \otimes \user2{r}_{3}] \otimes \user2{s}_1 \oplus \user2{f}_{9} \otimes \user2{s}_2 \oplus [\user2{f}_{2} \oplus \user2{f}_{5} \oplus \user2{f}_{3} \oplus \user2{f}_{3} \oplus \user2{f}_{6} \oplus \user2{f}_{7}] \otimes \user2{r}_{1} \otimes \user2{s}_3 \oplus \user2{f}_2 \otimes \user2{s}_4 \to \\ & \user2{f}_{8} \otimes \user2{r}_{1} \oplus \user2{f}_{1} \otimes \user2{r}_{2} \oplus \user2{f}_{9} \otimes \user2{r}_{3} \otimes \user2{s}_1 \oplus \user2{f}_{9} \otimes \user2{s}_2 \oplus [\user2{f}_{2} \oplus \user2{f}_{5} \oplus \user2{f}_{3} \oplus \user2{f}_{3} \oplus \user2{f}_{6} \oplus \user2{f}_{7}] \otimes \user2{r}_{1} \otimes \user2{s}_3 \oplus \user2{f}_2 \otimes \user2{s}_4 \mathop{\to}\limits^{(36)} \\ & \user2{f}_{8} \otimes \user2{r}_{1} \oplus \user2{f}_{1} \otimes \user2{r}_{2} \oplus [\user2{f}_{9} \otimes \user2{r}_{1} \oplus \user2{f}_{2} \otimes \user2{r}_{2} \oplus \user2{f}_{13} \otimes \user2{r}_{3} ] \otimes \user2{r}_{3} \otimes \user2{s}_1 \oplus \user2{f}_{13} \otimes \user2{s}_2 \oplus [\user2{f}_{2} \oplus \user2{f}_{5} \oplus \user2{f}_{3} \oplus \user2{f}_{3} \oplus \user2{f}_{6} \oplus \user2{f}_{7}] \otimes \user2{r}_{1} \otimes \user2{s}_3 \oplus \user2{f}_2 \otimes \user2{s}_4 \to \\ & \user2{f}_{8} \otimes \user2{r}_{1} \oplus \user2{f}_{1} \otimes \user2{r}_{2} \oplus \user2{f}_{9} \otimes \user2{r}_{1} \otimes \user2{r}_{3} \oplus \user2{f}_{2} \otimes \user2{r}_{2} \otimes \user2{r}_{3} \oplus \user2{f}_{13} \otimes \user2{r}_{3} \otimes \user2{r}_{3} \otimes \user2{s}_1 \oplus \user2{f}_{13} \otimes \user2{s}_2 \oplus [\user2{f}_{5} \oplus \user2{f}_{3} \oplus \user2{f}_{3} \oplus \user2{f}_{6} \oplus \user2{f}_{7}] \otimes \user2{r}_{1} \otimes \user2{s}_3 \oplus \user2{f}_5 \otimes \user2{s}_4 \mathop{\to}\limits^{(37)} \\ & \user2{f}_{8} \otimes\user2{r}_{1} \oplus \user2{f}_{1} \otimes \user2{r}_{2} \oplus\user2{f}_{9} \otimes \user2{r}_{1} \otimes \user2{r}_{3} \oplus \user2{f}_{2} \otimes \user2{r}_{2} \otimes \user2{r}_{3} \oplus [\user2{f}_{13} \otimes \user2{r}_{1} \oplus \user2{f}_{3} \otimes \user2{r}_{2} \oplus \user2{f}_{10} \otimes \user2{r}_{3}] \otimes \user2{r}_{3} \otimes \user2{r}_{3} \otimes \user2{s}_1 \oplus \user2{f}_{10} \otimes \user2{s}_2 \oplus \\ &[\user2{f}_{5} \oplus \user2{f}_{3} \oplus \user2{f}_{3} \oplus \user2{f}_{6} \oplus \user2{f}_{7}] \otimes \user2{r}_{1} \otimes \user2{s}_3 \oplus \user2{f}_5 \otimes \user2{s}_4 \\ \end{aligned} $$

Here, the parser expands the first two rules of grammar G 1 without any difficulty, since they are the same as the two first rules of the appropriate grammar G 2. After processing rule (37), IP → t I1, the first symbol on the input stack is the filler \(\user2{f}_5\) which represents the category DP3. However the upcoming filler to be handled is \(\user2{f}_3.\) At this point the parser breaks and does not reach the accepting final state by its trajectory. Similarly, another garden path occurs for processing the \({\tt so}\) sentence (33) according to grammar \(G_2(\tau_2({\tt so})),\) while \(\tau_2({\tt os})\) accepts the \({\tt os}\) sentence (33) as being well-formed regarding G 2 in the end.

Proof of the word semigroup representation theorem

What finally remains is to prove the assertion stated in section “Nonlinear dynamical automaton” that the quantum operators \(\rho(a_i), a_i \in{\mathbf{T}}\) provide a semigroup homomorphism and hence a representation of the word semigroup in the phase space X of an NDA (X, Φ). To this end, we have to generalize the definition Eqs. 60, 61 to proper words.

Let therefore \(u = u_{1}, \ldots, u_{p}, v = v_{1}, \ldots, v_{q}, w = w_{1}, \ldots, w_{r} \in{\mathbf{T}}^{\ast}\) be words over the terminal alphabet T of the NDA (X, Φ) with lengths \(|u| = p, |v| = q, |w| = r \in {\mathbb{N}}.\) As defined in Eq. 55, the Gödel code of the input word w of (X, Φ) with respect to the terminal number base b T is given as

$$ g_T(w) = \sum\limits_{i = 1}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} . $$

Call ρ′(u) the operator ρ (u) (Eq. 60) projected onto the y-component of the phase space. Its impact is then given by Eq. 61

$$ \rho^{\prime}(u)[g_T(w)] = g(w_1) b_T^{-1} + \sum\limits_{k = 1}^{|u|} g(u_k) b_T^{-k - 1} + \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-|u| - i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-|u| - i} , $$

which can be written as

$$ \rho^{\prime}(u)[g_T(w)] = g(w_1) b_T^{-1} + \tilde{g}_T(u) b_T^{-2} + \left[ \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} \right] b_T^{-|u|} $$

after introducing the function

$$ \tilde{g}_T(u) = \sum\limits_{k = 1}^{|u|} g(u_k) b_T^{-k + 1} . $$

For the proof we calculate

$$ \begin{aligned} &(\rho^{\prime}(u) \circ \rho^{\prime}(v))[g_T(w)] = \rho^{\prime}(u) \{ \rho^{\prime}(v) [g_T(w)] \}\\ =&\,\rho^{\prime}(u) \left\{ g(w_1) b_T^{-1} + \tilde{g}_T(v) b_T^{-2} + \left[ \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} \right] b_T^{-|v|} \right\} \\ =&\, g(w_1) b_T^{-1} + \tilde{g}_T(u) b_T^{-2} + \left\{ \tilde{g}_T(v) b_T^{-2} + \left[ \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} \right] b_T^{-|v|} \right\} b_T^{-|u|} \\ =&\, g(w_1) b_T^{-1} + \tilde{g}_T(u) b_T^{-2} + \tilde{g}_T(v) b_T^{-|u| - 2} + \left[ \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} \right] b_T^{-|u| - |v|} \\ =&\, g(w_1) b_T^{-1} + \left[ \tilde{g}_T(u) + \tilde{g}_T(v) b_T^{-|u|} \right] b_T^{-2} + \left[ \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} \right] b_T^{-|u \cdot v|} \\ =&\, g(w_1) b_T^{-1} + \tilde{g}_T(u \cdot v) b_T^{-2} + \left[ \sum\limits_{i = 2}^{|w|} g(w_i) b_T^{-i} + \sum\limits_{i = |w| + 1}^\infty g(\eta_{i - |w|}) b_T^{-i} \right] b_T^{-|u \cdot v|} \\ =&\, \rho^{\prime}(u \cdot v)[g_T(w)] . \\ \end{aligned} $$

QED.

Rights and permissions

Reprints and permissions

About this article

Cite this article

beim Graben, P., Gerth, S. & Vasishth, S. Towards dynamical system models of language-related brain potentials. Cogn Neurodyn 2, 229–255 (2008). https://doi.org/10.1007/s11571-008-9041-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11571-008-9041-5

Keywords

Navigation