"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

1. The after-effects of repetitive stimulation of the perforant path fibres to the dentate area of the hippocampal formation have been examined with extracellular micro-electrodes in rabbits anaesthetized with urethane.2. In fifteen out of eighteen rabbits the population response recorded from granule cells in the dentate area to single perforant path volleys was potentiated for periods ranging from 30 min to 10 hr after one or more conditioning trains at 10-20/sec for 10-15 sec, or 100/sec for 3-4 sec.3. The population response was analysed in terms of three parameters: the amplitude of the population excitatory post-synaptic potential (e.p.s.p.), signalling the depolarization of the granule cells, and the amplitude and latency of the population spike, signalling the discharge of the granule cells.4. All three parameters were potentiated in 29% of the experiments; in other experiments in which long term changes occurred, potentiation was confined to one or two of the three parameters. A reduction in the latency of the population spike was the commonest sign of potentiation, occurring in 57% of all experiments. The amplitude of the population e.p.s.p. was increased in 43%, and of the population spike in 40%, of all experiments.5. During conditioning at 10-20/sec there was massive potentiation of the population spike (;frequency potentiation'). The spike was suppressed during stimulation at 100/sec. Both frequencies produced long-term potentiation.6. The results suggest that two independent mechanisms are responsible for long-lasting potentiation: (a) an increase in the efficiency of synaptic transmission at the perforant path synapses; (b) an increase in the excitability of the granule cell population.

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Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit following stimulation of the perforant path.

This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

/pdf/dynamical-systems-in-neuroscience-51dkewn57v.pdf

Dynamical Systems in Neuroscience

Theoretical neuroscience provides a quantitative basis for describing what nervous systems do, determining how they function, and uncovering the general principles by which they operate This text introduces the basic mathematical and computational methods of theoretical neuroscience and presents applications in a variety of areas including vision, sensory-motor integration, development, learning, and memory The book is divided into three parts Part I discusses the relationship between sensory stimuli and neural responses, focusing on the representation of information by the spiking activity of neurons Part II discusses the modeling of neurons and neural circuits on the basis of cellular and synaptic biophysics Part III analyzes the role of plasticity in development and learning An appendix covers the mathematical methods used, and exercises are available on the book's Web site

Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems

Introduction to synaptic circuits, Gordon M.Shepherd and Christof Koch membrane properties and neurotransmitter actions, David A.McCormick peripheral ganglia, Paul R.Adams and Christof Koch spinal cord - ventral horn, Robert E.Burke olfactory bulb, Gordon M.Shepherd, and Charles A.Greer retina, Peter Sterling cerebellum, Rodolfo R.Llinas and Kerry D.Walton thalamus, S.Murray Sherman and Christof Koch basal ganglia, Charles J.Wilson olfactory cortex, Lewis B.Haberly hippocampus, Thomas H.Brown and Anthony M.Zador neocortex, Rodney J.Douglas and Kevan A.C.Martin Gordon M.Shepherd. Appendix: Dendretic electrotonus and synaptic integration.

The synaptic organization of the brain

A process and apparatus for the conversion of anilinothiazolino amino acid derivatives of peptides and proteins for use with conventional sequencers in the overall Edman degradation process. The apparatus comprises a modular unit separate from the sequencer having an independently programmable control and flow actuating and directing means including a source of pressurized nitrogen, a conversion reagent container, a wash solvent container, a reaction vessel, and a plurality of interconnecting tubes and valves controlled so as to regulate the amounts and durations of all fluids entering and leaving said reaction vessel under gas pressure.

Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input.

Branching dendritic trees and motoneuron membrane resistivity.

The sections in this article are:


1
Introduction
1.1
Core Conductor Concept

1.2
Perspective

1.3
Comment

1.4
Reviews and Monographs




2
Brief Historical Notes
2.1
Early Electrophysiology

2.2
Electrotonus

2.3
Passive Membrane Electrotonus

2.4
Passive Versus Active Membrane

2.5
Cable Theory

2.6
Core Conductor Concept

2.7
Core Conductor Theory

2.8
Estimation of Membrane Capacitance

2.9
Resting Membrane Resistivitiy

2.10
Passive Cable Parameters of Invertebrate Axons

2.11
Importance of Single Axon Preparations

2.12
Estimation of Parameters for Myelinated Axons

2.13
Space and Voltage Clamp




3
Dendritic Aspects of Neurons
3.1
Axon-Dendrite Contrast

3.2
Microelectrodes in Motoneurons

3.3
Theoretical Neuron Models and Parameters

3.4
Class of Trees Equivalent to Cylinders

3.5
Motoneuron Membrane Resistivity and Dendritic Dominance

3.6
Dendritic Electrotonic Length

3.7
Membrane Potential Transients and Time Constants

3.8
Spatiotemporal Effects with Dendritic Synapses

3.9
Excitatory Postsynaptic Potential Shape Index Loci

3.10
Comments on Extracellular Potentials

3.11
Additional Comments and References




4
Cable Equations Defined
4.1
Usual Cable Equation

4.2
Steady-state Cable Equations

4.3
Augmented Cable Equations

4.4
Comment: Cable Versus Wave Equation

4.5
Modified Cable Equation for Tapering Core

4.6
General Solution of Steady-state Cable Equation

4.7
Basic Transient Solutions of Cable Equation

4.8
Solutions Using Separation of Variables

4.9
Fundamental Solution for Instantaneous Point Charge




5
List of Symbols

6
Assumptions and Derivation of Cable Theory
6.1
One Dimensional in Space

6.2
Intracellular Core Resistance

6.3
Ohm's Law for Core Current

6.4
Conservation of Current

6.5
Relation of Membrane Current to Vi

6.6
Effect of Assuming Extracellular Isopotentiality

6.7
Passive Membrane Model

6.8
Resulting Cable Equation for Simple Case

6.9
Physical Meaning of Cable Equation Terms

6.10
Physical Meaning of τ

6.11
Physical Meaning of λ

6.12
Electrotonic Distance, Length, and Decrement

6.13
Effect of Placing Axon in Oil

6.14
Effect of Applied Current

6.15
Comment on Sign Conventions

6.16
Effect of Synaptic Membrane Conductance

6.17
Effect of Active Membrane Properties




7
Input Resistance and Steady Decrement with Distance
7.1
Note on Correspondence with Experiment

7.2
Cable of Semi-infinite Length

7.3
Comments about R∞, G∞, Core Current, and Input Current

7.4
Doubly Infinite Length

7.5
Case of Voltage Clamps at X1 and X2

7.6
Relations Between Axon Parameters

7.7
Finite Length: Effect of Boundary Condition at X= X1

7.8
Sealed End at X= X1: Case of B1 = 0

7.9
Voltage Clamp(V1 = 0) at X = X1: Case of B1 = ∞

7.10
Semi-infinite Extension at X = X1: Case of B1 = 1

7.11
Input Conductance for Finite Length General Case

7.12
Branches at X = X1

7.13
Comment on Branching Equivalent to a Cylinder

7.14
Comment on Membrane Injury at X = X1

7.15
Comment on Steady Synaptic Input at X= X1

7.16
Case of Input to One Branch of Dendritic Neuron Model




8
Passive Membrane Potential Transients and Time Constants
8.1
Passive Decay Transients

8.2
Time Constant Ratios and Electrotonic Length

8.3
Effect of Large L and Infinite L

8.4
Transient Response to Applied Current Step, for Finite Length

8.5
Applied Current Step with L Large or Infinite

8.6
Voltage Clamp at X = 0, with Infinite L

8.7
Voltage Clamp with Finite Length

8.8
Transient Response to Current Injected at One Branch of Model




9
Relations Between Neuron Model Parameters
9.1
Input Resistance and Membrane Resistivity

9.2
Dendritic Tree Input Resistance and Membrane Resistivity

9.3
Results for Trees Equivalent to Cylinders

9.4
Result for Neuron Equivalent to Cylinder

9.5
Estimation of Motoneuron Parameters

Core Conductor Theory and Cable Properties of Neurons

THE ORIGINAL OBJECTIVE of this research was to apply a mathematical theory of generalized dendritic neurons (36-39) to the interpretation and reconstruction of field potentials observed in the olfactory bulb of rabbit (32, 33). In the course of pursuing this objective, we were led to postulate dendrodendritic synaptic interactions which probably play an important role in sensory discrimination and adaptation in the olfactory system (40). More specifically, our initial aim was to develop a computational model, based on the known anatomical organization of the olfactory bulb and on generally accepted properties of nerve membrane, that could reconstruct the distribution of electric potential as a function of two variables, time and depth in the bulbar layers, following a synchronous antidromic volley in the lateral olfactory tract. The experimental studies of Phillips, Powell, and Shepherd (32, 33) had previously established that the recorded potentials at successive bulbar depths are highly reproducible and correlated with the histological layers of the bulb. These authors recognized the importance of this finding in relation to the symmetry and synchrony of activity in the mitral cell population; they deferred the interpretation of these potentials, in terms of specific neuronal activity, with a view to the present theoretical study.

Wilfrid Rall

Papers

Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input.

Branching dendritic trees and motoneuron membrane resistivity.

Core Conductor Theory and Cable Properties of Neurons

Theoretical reconstruction of field potentials and dendrodendritic synaptic interactions in olfactory bulb.

Dendrodendritic synaptic pathway for inhibition in the olfactory bulb