title: Interacting Turing-Hopf Instabilities Drive Symmetry-Breaking Transitions in a Mean-Field Model of the Cortex: A Mechanism for the Slow Oscillation

authors: Steyn-Ross, Moira L.

subjects: biological physics

abstract: Electrical recordings of brain activity during the transition from wake to anesthetic coma show temporal and spectral alterations that are correlated with gross changes in the underlying brain state. Entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (≲1  Hz) similar to the slow waves observed in natural sleep. Here we present a two-dimensional mean-field model of the cortex in which slow spatiotemporal oscillations arise spontaneously through a Turing (spatial) symmetry-breaking bifurcation that is modulated by a Hopf (temporal) instability. In our model, populations of neurons are densely interlinked by chemical synapses, and by interneuronal gap junctions represented as an inhibitory diffusive coupling. To demonstrate cortical behavior over a wide range of distinct brain states, we explore model dynamics in the vicinity of a general-anesthetic-induced transition from “wake” to “coma.” In this region, the system is poised at a codimension-2 point where competing Turing and Hopf instabilities coexist. We model anesthesia as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic response, producing a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space. The effect of long-range axonal white-matter connectivity is probed with the inclusion of a single idealized point-to-point connection. We find that the additional excitation from the long-range connection can provoke seizurelike bursts of cortical activity when inhibitory diffusion is weak, but has little impact on an active cortex. Our proposed dynamic mechanism for the origin of anesthetic slow waves complements—and contrasts with—conventional explanations that require cyclic modulation of ion-channel conductances. We postulate that a similar bifurcation mechanism might underpin the slow waves of natural sleep and comment on the possible consequences of chaotic dynamics for memory processing and learning.

text: Interacting Turing-Hopf Instabilities Drive Symmetry-Breaking Transitions in a Mean-Field Model of the Cortex: A Mechanism for the Slow Oscillation Electrical recordings of brain activity during the transition from wake to anesthetic coma show temporal and spectral alterations that are correlated with gross changes in the underlying brain state. Entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (≲1  Hz) similar to the slow waves observed in natural sleep. Here we present a two-dimensional mean-field model of the cortex in which slow spatiotemporal oscillations arise spontaneously through a Turing (spatial) symmetry-breaking bifurcation that is modulated by a Hopf (temporal) instability. In our model, populations of neurons are densely interlinked by chemical synapses, and by interneuronal gap junctions represented as an inhibitory diffusive coupling. To demonstrate cortical behavior over a wide range of distinct brain states, we explore model dynamics in the vicinity of a general-anesthetic-induced transition from “wake” to “coma.” In this region, the system is poised at a codimension-2 point where competing Turing and Hopf instabilities coexist. We model anesthesia as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic response, producing a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space. The effect of long-range axonal white-matter connectivity is probed with the inclusion of a single idealized point-to-point connection. We find that the additional excitation from the long-range connection can provoke seizurelike bursts of cortical activity when inhibitory diffusion is weak, but has little impact on an active cortex. Our proposed dynamic mechanism for the origin of anesthetic slow waves complements—and contrasts with—conventional explanations that require cyclic modulation of ion-channel conductances. We postulate that a similar bifurcation mechanism might underpin the slow waves of natural sleep and comment on the possible consequences of chaotic dynamics for memory processing and learning.

year: 2013

title: A similarity reduction of the Grad-Shafranov equation

authors: Litvinenko, Yuri E.

subjects: mathematics

abstract: A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad–Shafranov equation. As an illustration of the method, the frequently used Solov’ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad–Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions.

text: A similarity reduction of the Grad-Shafranov equation A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad–Shafranov equation. As an illustration of the method, the frequently used Solov’ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad–Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions.

year: 2010

title: Intersecting knowledge worlds: Describing the cultural interfaces within a socioscientific issue

authors: Wagner, Brent

subjects: cultural interface

abstract: Increasing numbers of science teachers in Aotearoa New Zealand are exploring ways to incorporate indigenous mātauranga Māori (Māori knowledge and ways of knowing) into their science programmes. This article presents findings from a study that examined how senior biology students in two English-medium secondary schools incorporated mātauranga Māori alongside biological science knowledge in their discussions about the use of 1080 poison to control mammalian predators. From a social constructivist ontology and interpretivist mode of inquiry using qualitative methodology, a triadic epistemic agency conceptual framework was developed inductively and used to analyse students’ written scripts for their national school leaving qualification (NCEA), and transcriptions of individual interviews. The study took place prior to the introduction of equal status to mātauranga Māori within the New Zealand Curriculum. Deductive analysis using the Triadic identified a group of ten students who drew from both knowledge systems, with varying degrees of knowledge integration. In addition to detailed representations of the theoretical space of the locale of the learner, the findings also identified three distinctive cultural interfaces occupied by the students: conflicted, parallel, and, connected. The findings underscore the importance of enabling diverse knowledge systems in biology classes to converge. Specifically, the incorporation of mātauranga Māori alongside biological science knowledge encourages a critical stance towards knowledge claims, supports pluralist knowledge use among culturally and linguistically diverse learners and promotes critical engagement with socioscientific issues (SSI) in senior secondary school biology.

text: Intersecting knowledge worlds: Describing the cultural interfaces within a socioscientific issue Increasing numbers of science teachers in Aotearoa New Zealand are exploring ways to incorporate indigenous mātauranga Māori (Māori knowledge and ways of knowing) into their science programmes. This article presents findings from a study that examined how senior biology students in two English-medium secondary schools incorporated mātauranga Māori alongside biological science knowledge in their discussions about the use of 1080 poison to control mammalian predators. From a social constructivist ontology and interpretivist mode of inquiry using qualitative methodology, a triadic epistemic agency conceptual framework was developed inductively and used to analyse students’ written scripts for their national school leaving qualification (NCEA), and transcriptions of individual interviews. The study took place prior to the introduction of equal status to mātauranga Māori within the New Zealand Curriculum. Deductive analysis using the Triadic identified a group of ten students who drew from both knowledge systems, with varying degrees of knowledge integration. In addition to detailed representations of the theoretical space of the locale of the learner, the findings also identified three distinctive cultural interfaces occupied by the students: conflicted, parallel, and, connected. The findings underscore the importance of enabling diverse knowledge systems in biology classes to converge. Specifically, the incorporation of mātauranga Māori alongside biological science knowledge encourages a critical stance towards knowledge claims, supports pluralist knowledge use among culturally and linguistically diverse learners and promotes critical engagement with socioscientific issues (SSI) in senior secondary school biology.

year: 2026