Cognitive Neurodynamics

, Volume 2, Issue 3, pp 229–255

Towards dynamical system models of language-related brain potentials

Authors

    • School of Psychology and Clinical Language SciencesUniversity of Reading
  • Sabrina Gerth
    • Institute for LinguisticsUniversity of Potsdam
  • Shravan Vasishth
    • Institute for LinguisticsUniversity of Potsdam
Research Article

DOI: 10.1007/s11571-008-9041-5

Cite this article as:
beim Graben, P., Gerth, S. & Vasishth, S. Cogn Neurodyn (2008) 2: 229. doi:10.1007/s11571-008-9041-5

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.

Keywords

Computational psycholinguisticsLanguage processingEvent-related brain potentialsDynamical systems

Supplementary material

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

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