Piecewise linear neuron modeling

a neuron modeling and piecewise technology, applied in the field of artificial nervous systems, can solve problems such as inability to find exact solutions, and achieve the effect of accurate modeling of dynamics and low complexity

Inactive Publication Date: 2014-05-22
QUALCOMM INC
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  • Description
  • Claims
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AI Technical Summary

Benefits of technology

[0007]Certain aspects of the present disclosure generally relate to approximating at least a portion of a nonlinear function of a neuron model as a piecewise linear function. Methods and apparatus for implementing the resulting linearized neuron model in one or more artificial neurons, for example, are also provided. Certain aspects of the present disclosure generally relate to a common and flexible architecture for the implementation of the dynamics of neuron models. The design goals include low complexity, accurate modeling of the dynamics, and the ability to implement any neuron model (of one, two, or more dimensions). The piecewise linear approximations provide a simple way to change neuron models within such an architecture, simply by substituting different parameters associated with various neuron models.

Problems solved by technology

Since the system is nonlinear, exact solutions cannot be found.

Method used

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Examples

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example 1

Taylor Expansion for the Izhikevich (Simple) Model

[0164]In this example, Eq. (68) is evaluated for the simple model and linearization based on the Taylor expansion method. To simplify the formulas somewhat, it is assumed that the external current is not present.

[0165]First, the coefficients of the Taylor expansion for the simple model, i.e., F(v)=k(v−vr)(v−vt), are derived starting from Eq. (56). Provided that

1C∂F(v)∂v|vn=2kCvn-kC(vr+vt)(69)

one obtains

a11[vn,nT]=2kCvn-kC(vr+vt)+1C∑i=1Lgi(nT)[hi′(vn)(Ei-vn)-hi(vn)](70)b1[vn,t]=Γ(vn,t)-a11[vn,nT]vn=1C{k(vn-vr)(vn-vt)+∑i=1Lgi(t)hi(vn)(Ei-vn)}-{2kCvn2-kC(vr+vt)vn+1C∑i=1Lgi(nT)[hi′(vn)(Ei-vn)-hi(vn)]vn}(71)

[0166]For non-NMDA synaptic channels, Eqs. (70) and (71) simplify to

a11[vn,nT]=2kCvn-kC(vr+vt)-1C∑i=1Lgi(nT)(72)b1[vn,nT]=1C{-kvn2+kvrvt+∑i=1Lgi(nT)vn}+1C∑i=1Lgi(t)(Ei-vn)(73)

[0167]Taking

b0=1C{-kvn2+kvrvt+∑i=1Lgi(nT)[hi′(vn)(Ei-vn)-hi(vn)]vn}

Eq. (68) now becomes

x(nT+T)=AnTx(nT)+An-1(AnT-I)[b0b2]+∑i=1L[hi(vn)(Ei-vn)]∫0TAnτgi(nT+T-τ)τ[10...

example 2

Subthreshold Dynamics of the Hunzinger Cold Model

[0171]Another example is developed in an effort to examine the subthreshold dynamics of the simple intrinsic conductance model known as the Hunzinger Cold model. In this example, no synaptic currents and the simple, but interesting case of impulsive external current are assumed.

[0172]In the Hunzinger Cold model (see Eq. (22) above) when the membrane voltage is below threshold, the matrix, An is constant and equal to A−:

A-≡[-1 / τ--1 / Cab-a]

[0173]The derivation can be further simplified if the first state variable is defined as the membrane voltage minus the reference voltage. With such a definition, all the constant terms are equal to zero, and Eq. (61) simplifies to

x(t)=A-(t-Tn)x(Tn)+1C∫TntA-(t-τ)Iext(τ)τ[10](77)

[0174]Furthermore, if the external current is assumed to be a Dirac delta function at time Tn with amplitude I, i.e., Iext=(t)+Iδ(t−Tn), then

x(t)=A-(t-Tn)(x(Tn)+1C[10])(78)

[0175]Note that impulsive inputs have the same effect on...

example 3

Approximation with a Backward Rectangular Rule

[0189]To simplify the exposition and notations, this example assumes that the synaptic current does not contain voltage-dependent conductance channels (e.g., NMDA channels) and that there is no external current Iext(t). The derivation is obtained for fixed step sizes of length T. To begin, Eq. (68) may be simplified as follows:

x(nT+T)=AnTx(nT)+q+1C∫0TAnτ∑i=1Lgi(nT+T-τ)(Ei-v(nT+T))τ[10](98)

where the vector q contains constant terms, namely

q=An-1(AnT-I)[b0b2](99)

[0190]For notational convenience, the following vector and matrix may also be defined:

b≡[10]B≡[1000](100)

such that Eq. (98) may be rewritten as

x(nT+T)=AnTx(nT)+1C∫0TAnT∑i=1Lgi(nT+T-τ)Eiτb-1C∫0TAnτ∑i=1Lgi(nT+T-τ)τB·x(nT+T)+q(101)

[0191]Now, the solution is derived for gi(t) modeled as simple exponentials per Eq. (29). In this case, one has

x(nT+T)=AnTx(nT)+1C∑i=1Lgi(nT)Ei∫0TAnτ-(T-τ) / τiτ·b-1C∑i=1Lgi(nT)∫0TAnτ-(T-τ) / τiτB·x(nT+T)+q(102)

[0192]The integrals can be readily solved, and if o...

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Abstract

Methods and apparatus for piecewise linear neuron modeling and implementing one or more artificial neurons in an artificial nervous system based on one or more linearized neuron models. One example method (for implementing a combination of a plurality of neuron models in a system of neural processing units) generally includes loading parameters for a first neuron model selected from the plurality of neuron models into a first neural processing unit, determining a first state of the first neural processing unit based at least in part on the parameters for the first neuron model, and determining a second state of the first neural processing unit based at least in part on the parameters for the first neuron model and on the first state. This method may also include updating the plurality of neuron models (e.g., by adding, deleting, or adjusting parameters for the first neuron model or another neuron model).

Description

CLAIM OF PRIORITY UNDER 35 U.S.C. §119[0001]This application claims benefit of U.S. Provisional Patent Application Ser. No. 61 / 728,360, filed Nov. 20, 2012 and entitled “Piecewise Linear Neuron Modeling,” U.S. Provisional Patent Application Ser. No. 61 / 734,716, filed Dec. 7, 2012 and entitled “Piecewise Linear Neuron Modeling,” U.S. Provisional Patent Application Ser. No. 61 / 740,633, filed Dec. 21, 2012 and entitled “Piecewise Linear Neuron Modeling,” and U.S. Provisional Patent Application Ser. No. 61 / 756,889, filed Jan. 25, 2013 and entitled “Piecewise Linear Neuron Modeling,” all of which are herein incorporated by reference in their entireties.BACKGROUND[0002]1. Field[0003]Certain aspects of the present disclosure generally relate to artificial nervous systems and, more particularly, to approximating at least a portion of a nonlinear function of a neuron model as a piecewise linear function and to using the resulting linearized neuron model in one or more artificial neurons.[000...

Claims

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Application Information

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Patent Type & Authority Applications(United States)
IPC IPC(8): G06N3/02
CPCG06N3/049G06N3/04G06N3/063G06N3/08G06N3/082G06N20/00
Inventor PADOVANI, ROBERTOYOON, YOUNG CHEUL
Owner QUALCOMM INC
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