Biological Cybernetics
Biol. Cybernetics 33, 237-247 (1979)
9 by Springer-Verlag 1979
Nonlinear Gain Mediating Cortical Stimulus-Response Relations* Walter J. Freeman Department of Physiology-Anatomy,Universityof California, Berkeley,California,USA
Abstract. Single isolated neurons show a nonlinear increase in likelihood of firing in response to random input, when they are biased toward threshold by steady-state depolarization. It is postulated that this property holds for neurons in the olfactory bulb, a specialized form of cortex in which the steady state is under centrifugal control. A model for this nonlinearity is based on two first order differential equations that interrelate three state variables: activity density in the pulse mode p of a local population (subset) of neurons ; activity density in the wave mode u (mean dendritic current); and an intervening variable m that may be thought to represent the average subthreshold value in the subset for the sodium activation factor as defined in the Hodgkin-Huxley equations. A stable steady state condition is posited at (P0,Uo, mo). It is assumed that: (a) the nonlinearity is static; (b) m increases exponentially with u; (c) p approaches a maximum Pm asymptotically as m increases; (d) p > 0; and (e) the steady state values of Po, Uo, and m o are linearly proportional to each other. The model is tested and evaluated with data from unit and E E G recording in the olfactory bulb of anesthetized and waking animals. It has the following properties. (a) There is a small-signal near-linear input-output range. (b) There is bilateral saturation for large input, i.e. the gain defined as dp/du approaches zero for large lu]. (c) The asymptotes for large input + u are asymmetric. (d) The range of output is variable depending on u o. (e) Most importantly, the maximal gain occurs for u > u0, so that positive (excitatory) input increases the output and also the gain in a nonlinear manner. It is concluded that large numbers of neurons in the olfactory
* Supported by a grant MH 06686 from the National Institute of Mental Health, and by a Research Professorship from the Miller Institute The author is gratefulto Dr. Daniel M. Sunday,Dr. Soo-Myung Ahn and Brian C. Burke for assistance in various aspects of this work
bulb of the waking animal are maintained in a sensitive nonlinear state, that corresponds to the domain of the subthreshold local response of single axons, as it is defined by Rushton and Hodgkin.
Introduction
Sensory systems are remarkable for their combined sensitivity and stability. A stimulus that is sufficient to excite one or a few receptors in a prepared animal can give rise to widespread autonomic, postural and purposive locomotor changes within 200-300ms, which implies that the initial activity has been rapidly spread widely through the forebrain in a controlled manner. It further suggests the existence at an early stage in each sensory system of neural mechanisms for rapid increase in the numbers of neurons responding to a stimulus from one or a few neurons to many, say 104 or more, as the basis for dissemination through the CNS. One plausible mechanism requires that receptor input to a small number of 2nd order sensory neurons not only excite them but increase their transmission strength. If these neurons were excitatory to each other as well as others in their neighborhood, then regenerative neural activity would result. That is, the stimulus action would be to increase the feedback gain in a local ensemble of neurons that was prepared to receive the stimulus. But the ensemble must be stable in three respects. The response should not occur or should be distinctly different in the presence of adventitious events (noise). It should be self-limiting in its maximal amplitudes. And it should be rapidly selfterminating, so that a new stimulus sample can be taken through the same ensemble within a few hundred ms. The key to understanding this mechanism is the nonlinear relationship between input and output for a 0340-1200/79/0033/0237/$02.20
238
L~
-2
-~
l AVE AFIPLITUDE
7
3
-3
2
-I 1 WAVE AMPLITUDE
2
3
3 WAVE AMPLITUDE
Fig. 1. Three examples are shown of the normalized conditional pulse probability distribution (A) and the numerically computed pulse probability density (~) from anesthetized rabbits. The curves fitted to the pulse probability distribution (A) are from Eqs. (5.0), (5.1), (10.0), and (10.1). The parameters are: left middle right
Q~ a Ap Av 1.00 0.10 -0.16 -0.16 2.02 0.55 0.52 0.42 2.33 1.10 0.09 0.09.
The experimental origin is at (P(f/)- ~, ~ - u). The small cross in each frame shows the estimated true origin from evaluation of Av and Ap
subset of neurons in parallel within an ensemble (Freeman, 1975). Each neuron performs two timedependent nonlinear operations. Incoming axonal impulse trains (the "pulse" mode) are converted to continuously varying dendritic current (the "wave" mode) at synapses, and the current serves to control the generation of impulses at trigger zones. The distributions of pulses for single neurons tend to be Poisson, and the wave amplitude distributions to be Gaussian. Over a subset of neurons, in which the time scale of a sensory event is one or two orders of magnitude longer than the time scale of a nerve impulse, the pulses give rise to a continuous pulse density, and the pulse-wave and wave-pulse transformations can be treated as static nonlinearities (Baker, 1978). The input-output operation for the subset can be expressed as the product of two nonlinear functions in pulse and wave amplitude. The derivative of the nonlinear function with respect to wave amplitude gives the nonlinear gain of the operation. An expression for the nonlinear function has been derived experimentally (Freeman, 1975) from the relation of cortical neural firing probabilities to the EEG. The frequency of firing of a neuron or a cluster of neurons recorded with a microelectrode is taken as a manifestation of the pulse density of a neural subset around the electrode. In the appropriate neural geometry (Freeman, 1975, p. 233) the E E G recorded at or near the subset is a manifestation of the average membrane potential of neurons in a subset. A set of simultaneous measurements of E E G amplitude and pulse occurrence sampled at a high rate (e.g. 1 ms) over an extended time span (e.g. 100s) yields an E E G
amplitude histogram and the mean pulse rate. Then the empirical probability of cell firing is determined conditional on the E E G amplitude. This pulse probability distribution is an expression of the nonlinear operations of pulse-wave or wave-pulse conversion. Its numerically computed derivative with respect to wave amplitude is a pulse probability density 1 that is equivalent to the nonlinear gain. This gain curve is the focus of interest in this report.
Conditional Pulse Probability Distribution and Density The nonlinear function has been derived experimentally by analysis of the relation of nerve impulses to the E E G in the olfactory system of cats and rabbits. Details have already been published (Freeman, 1975). Units were recorded singly or in clusters from a microelectrode in cortex p(t) while the E E G was recorded from a surface electrode v(t). After appropriate amplification and filtering of the two signals, the experimental pulse probability conditional on E E G amplitude and time lag was calculated as follows. The E E G was band-pass filtered to remove unit activity and the low-frequency respiratory wave of the olfactory E E G and was then digitized over a time segment 1 Confusion should be avoided between pulse density p and conditional pulse probability density dP(V)/dV The pulse density denotes the pulse rate/unit volume of tissue enclosing a neural subset. It can be estimated by ensembleaveraging over unit trains as in constructing a post-stimulus time histogram. The pulse probability density is estimated by constructing the pulse probability distribution conditional on amplitude V and then computing the derivative with respect to V
239 1. 0
-i.
3
-7
-i WAVE
0 I AMPLITUDE
7
3
~'~-3
-7
-I WAVE
0 i AMPI_ITUDE
2
3 WAVE
AMPLI~UOE
Fig. 2. Three examplesare shown of the normalizedconditional pulse probabilitydistribution (z~)and density([B)fromwakingrabbits; no clear upper asymptotes are seen. The fitted curves (see legend for Fig. 1) have the parameters:
left middle right
Q. 4.84 6.83 14.86
cr 0,33 0.56 0.52.
Ap 0.97 0.91 1.62
Av 0.77 0.74 1.05
of 10-100s; the N values were normalized to zero mean and unit standard deviation, a. The amplitude range was divided into 60 bins 0.1 in width from + 3 to - 3. For each occurrence of an amplitude V at time t the question was asked, did a pulse occur at each time lag T in p(t + T), - 25 ms < T < + 25 ms ? If so, a count of 1 was added to the appropriate bin of a table over and T. The resulting histogram was divided by N and divided by the amplitude probability density to give the conditional pulse probability/5(f/,, T). Cross sections of the table at fixed ~'+ 0 gave/3(T), which was oscillatory at the frequency of the EEG. Cross sections a t T = Tmax where the value of P(T) was maximal gave P(V), which was normalized by dividing the function by the mean firing rate ~. The main conclusions from analysis of experimental curves from the olfactory bulb in anesthetized animals were as follows. a) The maximally varying nonlinear functions were found at Tmax values for which the average of Tmax was equal to one quarter cycle of the dominant frequency of the EEG. This reflected the fact that the E E G was generated by inhibitory neurons and the pulses by excitatory neurons forming a negative feedback loop in the bulb. b) For low amplitude EEG activity/3(f/) at T = Tin,X was nearly linear over the interval + 3o-. c) During high amplitude EEG activity the curves became nonlinear with (on the average) monotonically increasing pulse probability P with increasing wave amplitude and horizontal asymptotes (a "sigmoid" curve as shown in Fig. 1). The sigmoid curve was interpreted to show wave-pulse conversion (wave being the independent variable).
d) The sigmoidal curves were asymmetric about the mean firing rate ~, with a lower asymptote at/3 = 0 and an upper asymptote at 3Po. Analysis of the numerically computed derivatives of the experimental curves (Sunday and Freeman, 1975) showed that e) the dominant peak of the pulse probability density usually occurred in the interval V=(a/2, a), and its magnitude increased linearly with ~. f) The experimental curves from waking rabbits, in which EEG amplitudes were much greater than in anesthetized animals, revealed with increasing EEG amplitude a marked increase in the steepness of the nonlinear curve, the degree of asymptotic asymmetry, and the displacement of the peak of the density curve toward higher amplitude (Fig. 2). The newer findings e)-f) were incompatible with an earlier model (Freeman, 1975), in which it was assumed that the maximal nonlinear gain was at v = 0 , and that the degree of asymptotic asymmetry was fixed at 2, so the construction of a new model was undertaken.
Derivation of the Model
The element of cortical function is taken to be an ensemble of neurons in a local neighborhood that can be lumped without regard to spatial dimensions. The ensemble is described in terms of its topology (the network of connections) and the operations performed by the subset of neurons at each node in the network. The operations of the subset occur in parallel at the same time and are the statistical averages of the operations of the neurons comprising the subset. This report deals with the operations; the t o p o l o g y need not be considered here.
240 M
/3
In the steady state wherein dp/dt = 0 and du/dt = 0, m = m o, u=u o and P=Po. The variables for computation and experimental measurement are
P
v ~- f l ( u -
uo),
q&P-Po,
t.~ c:3
Po
I O
I
i
(1.1)
where & means "is defined as equal to" and fl is a scaling factor. The dimensionless normalized variables are
z Ii20
(1.0)
0
V
u
WAVE AMPLITUDE
Fig. 3. Normalizedpulse densityQ is a functionof wave amplitude u. The steady state values are u=u o and P=Po. The normalized variables.are q =P-Po and v=u-u o. The interveningvariable m is orthogonal to p and u but is shown in the p-u plane for convenience. Curve M is from Eq. (2.2). Curve Q is from Eq. (5.0)
Q & (P- Po)/Po, m & ( m - mo)/m o .
G,(v)F ~(p)Ga(p)F d(v) . When, in the physiological range of operation, the axonal nonlinearity G,(v) dominates and thereby restricts the dendritic operation Ga(p) to a linear range, Gd(p) can be replaced by a constant (. Then the two linear operations can be combined into a single operator F(p) in the sequence
G~(v)(F(p). The linear operation (F(p) need not be considered further here. The proposed model of the nonlinearity contains three state variables. The first two p and u (Fig. 3) represent activity density respectively in the pulse and wave modes. Alternatively q and v are normalized from p and u to zero in the steady state and are used for observation, measurement and computation. The third variable m is to be defined.
(1.3)
Three properties are assigned to the neural subset in the local ensemble. First it is assumed that the m variable is regeneratively voltage-dependent. Corresponding to this property,
dm = c~m. du
(2.0)
For any change in u from u o and m from m o u
=~ j du. Each subset has two state variables : activity density in the pulse density mode p(t) in axons and activity density in the wave mode v(t) in dendrites. There are four operations. Dendritic activity is converted to axonal activity at trigger zones by a nonlinear operation Ga(v). It is linearly delayed and dispersed in time by axonal propagation and multiplied by axonal branching Fa(p). Axonal activity is re-converted to dendritic activity at synapses by a nonlinear operation Ga(p). The synaptically induced activity is integrated and delivered to the trigger zones by a linear operation Fd(v ). In the order of operation the sequence is
(1.2)
(2.1)
UO
From Eqs. (1.0) and (1.3) M = e ~ - 1.
(2.2)
Second, it is assumed that p increases with increasing m, but that there exists an upper limit on pulse density p,, for any given steady state P0 and u o. There are many factors in this limitation, some (such as voltage-dependent saturation of Na-activation, potassium conductance and sodium inactivation) depending on membrane potential, others (such as hyperpolarizing after-potentials) depending on previous activity, and these factors are expressions of nonlinear and time-varying relationships. Over the subset and over the longer time scale of its operation (compared to the time scale of pulses) it is assumed that the nonlinearities become static, and that the rate of increase in pulse density dp/dm is proportional to the difference between p and the prevailing maximum p,, and is inversely proportional to the range P0-P,,.
dp _fl, p~,-p dm P,,- Po'
(3.0)
where fi' is a scaling factor. Integration from Po to p and from m o to m,
fl' " ~voP~--P Pm~PoJo dm' P
dp
_
(3.1)
gives
p=pO+(pm--Po)(1--e--e'(m--mo)/(Pm--Po)).
(3.2)
241 To simplify the notation the prevailing maximal pulse density Pm is expressed as a dimensionless number from Eq. (1.2). (3.3)
Qm ~ (pro- Po)/po .
From Eqs. (1.1), (1.3) and (3.2) Q=Q.,ntl - e
(3.4)
Qmno).
The third property of the system is taken from the empirical observation that for most neurons over a physiological range the steady state pulse rate is proportional to steady state transmembrane current or level of depolarization (Granit et al., 1963 ; Stein, 1967; Calvin, 1975; Heyer and LlinAs, 1976). Po =~Uo,
(4.0)
where 7 is a scaling factor. This linear relationship cannot be derived from the Hodgkin-Huxley system (Agin, 1964 ; Cooley and Dodge, 1966), although it can be extracted by appropriate modifications of the system (Dodge, 1972; Fohlmeister et al., 1974). The relationships between m o and both Po and u o must also be specified. Intuitively it seems that m o should increase monotonically with u o over the physiological range of operation of a neural subset, and P0 should increase monotonically with mo. Equation (4.0) requires that each monotonic function be the inverse of the other. In the absence of experimental or theoretical evidence to the contrary, it is assumed that (4.1)
Po = fi'mo.
The derivative of Eq. (5.0) is equivalent to the experimental pulse probability density curve. It is found by treating the two implicit operations [(2.0) and (3.0)] as serial dQ
d M dQ
dv
dv dM"
(6.0)
From Eqs. (2.2) and (3.4)
dQ
dv = exp [v - (e~- 1)/Qm].
(6.1)
At V=Vo=0 , Qo = 0 ,
dQ dv
(6.2)
- 1.
(6.3)
Equation (6.1) describes the forward gain in the operation Ga(v ). The gain approaches zero as v approaches A _+ oo and goes to a maximum at v=%. The second derivative d~v dvv =
(1-e~/Qm)'
(6.4)
is zero at v = v0, so that (6.5)
v o= ln(Q,, ) .
The peak of the gain curve shifts to the right and its height increases with increased Qm and [-by Eq. (4.4)] u o. The third derivative is
Since m is not measured, fi'= 1. The relationship between u o and Qm is based on the empirical property that p > 0 and therefore Q > - 1. The value for u = 0 is set at Q = - 1 . F r o m Eqs. (1.0), (2.2), and (3.4)
&2 Td =
m = e-P'~ _ 1,
When v =0, there is a maximum in the second derivative for
(4.2)
1
e -MI~ = 1 -~- - Qm"
(4.3)
1
,44
On combining Eqs. (2.2), (3.4), (4.0), and (4.4) (see Fig. 3)
e v = Qm(1.5 + ]/5/2).
Q = - 1 , v = < - u o.
(5.0)
(6.7)
(6.8)
This value for Qm provides a convenient dividing point between high and low sensitivity. For Qm 0 is negative, whereas for Q,, > 2.62 the acceleration is positive. The nonlinear gain for wave-pulse conversion is dp
(2 = Qm(1 - exp[-- (e~ - 1)/Qm]) , v > - uo,
(6.6)
and is zero at
Q,, =2.62.
Because u o is not measured, fl = 1 and
uo ,hi,
(1- 3e~176
d~ = ?u~ exp(v - e - @- 1)/Qm),
(6.9)
and the maximal gain at v = vg is
From Eqs. (1.2) and (4.0) the pulse output is p =TUo( Q + 1).
(5.1)
dv o = T u ~
l + lle"
(6.10)
242 16, 14
12 z4
Biol. Cybernetics 33, 237-247 (1979)
9 by Springer-Verlag 1979
Nonlinear Gain Mediating Cortical Stimulus-Response Relations* Walter J. Freeman Department of Physiology-Anatomy,Universityof California, Berkeley,California,USA
Abstract. Single isolated neurons show a nonlinear increase in likelihood of firing in response to random input, when they are biased toward threshold by steady-state depolarization. It is postulated that this property holds for neurons in the olfactory bulb, a specialized form of cortex in which the steady state is under centrifugal control. A model for this nonlinearity is based on two first order differential equations that interrelate three state variables: activity density in the pulse mode p of a local population (subset) of neurons ; activity density in the wave mode u (mean dendritic current); and an intervening variable m that may be thought to represent the average subthreshold value in the subset for the sodium activation factor as defined in the Hodgkin-Huxley equations. A stable steady state condition is posited at (P0,Uo, mo). It is assumed that: (a) the nonlinearity is static; (b) m increases exponentially with u; (c) p approaches a maximum Pm asymptotically as m increases; (d) p > 0; and (e) the steady state values of Po, Uo, and m o are linearly proportional to each other. The model is tested and evaluated with data from unit and E E G recording in the olfactory bulb of anesthetized and waking animals. It has the following properties. (a) There is a small-signal near-linear input-output range. (b) There is bilateral saturation for large input, i.e. the gain defined as dp/du approaches zero for large lu]. (c) The asymptotes for large input + u are asymmetric. (d) The range of output is variable depending on u o. (e) Most importantly, the maximal gain occurs for u > u0, so that positive (excitatory) input increases the output and also the gain in a nonlinear manner. It is concluded that large numbers of neurons in the olfactory
* Supported by a grant MH 06686 from the National Institute of Mental Health, and by a Research Professorship from the Miller Institute The author is gratefulto Dr. Daniel M. Sunday,Dr. Soo-Myung Ahn and Brian C. Burke for assistance in various aspects of this work
bulb of the waking animal are maintained in a sensitive nonlinear state, that corresponds to the domain of the subthreshold local response of single axons, as it is defined by Rushton and Hodgkin.
Introduction
Sensory systems are remarkable for their combined sensitivity and stability. A stimulus that is sufficient to excite one or a few receptors in a prepared animal can give rise to widespread autonomic, postural and purposive locomotor changes within 200-300ms, which implies that the initial activity has been rapidly spread widely through the forebrain in a controlled manner. It further suggests the existence at an early stage in each sensory system of neural mechanisms for rapid increase in the numbers of neurons responding to a stimulus from one or a few neurons to many, say 104 or more, as the basis for dissemination through the CNS. One plausible mechanism requires that receptor input to a small number of 2nd order sensory neurons not only excite them but increase their transmission strength. If these neurons were excitatory to each other as well as others in their neighborhood, then regenerative neural activity would result. That is, the stimulus action would be to increase the feedback gain in a local ensemble of neurons that was prepared to receive the stimulus. But the ensemble must be stable in three respects. The response should not occur or should be distinctly different in the presence of adventitious events (noise). It should be self-limiting in its maximal amplitudes. And it should be rapidly selfterminating, so that a new stimulus sample can be taken through the same ensemble within a few hundred ms. The key to understanding this mechanism is the nonlinear relationship between input and output for a 0340-1200/79/0033/0237/$02.20
238
L~
-2
-~
l AVE AFIPLITUDE
7
3
-3
2
-I 1 WAVE AMPLITUDE
2
3
3 WAVE AMPLITUDE
Fig. 1. Three examples are shown of the normalized conditional pulse probability distribution (A) and the numerically computed pulse probability density (~) from anesthetized rabbits. The curves fitted to the pulse probability distribution (A) are from Eqs. (5.0), (5.1), (10.0), and (10.1). The parameters are: left middle right
Q~ a Ap Av 1.00 0.10 -0.16 -0.16 2.02 0.55 0.52 0.42 2.33 1.10 0.09 0.09.
The experimental origin is at (P(f/)- ~, ~ - u). The small cross in each frame shows the estimated true origin from evaluation of Av and Ap
subset of neurons in parallel within an ensemble (Freeman, 1975). Each neuron performs two timedependent nonlinear operations. Incoming axonal impulse trains (the "pulse" mode) are converted to continuously varying dendritic current (the "wave" mode) at synapses, and the current serves to control the generation of impulses at trigger zones. The distributions of pulses for single neurons tend to be Poisson, and the wave amplitude distributions to be Gaussian. Over a subset of neurons, in which the time scale of a sensory event is one or two orders of magnitude longer than the time scale of a nerve impulse, the pulses give rise to a continuous pulse density, and the pulse-wave and wave-pulse transformations can be treated as static nonlinearities (Baker, 1978). The input-output operation for the subset can be expressed as the product of two nonlinear functions in pulse and wave amplitude. The derivative of the nonlinear function with respect to wave amplitude gives the nonlinear gain of the operation. An expression for the nonlinear function has been derived experimentally (Freeman, 1975) from the relation of cortical neural firing probabilities to the EEG. The frequency of firing of a neuron or a cluster of neurons recorded with a microelectrode is taken as a manifestation of the pulse density of a neural subset around the electrode. In the appropriate neural geometry (Freeman, 1975, p. 233) the E E G recorded at or near the subset is a manifestation of the average membrane potential of neurons in a subset. A set of simultaneous measurements of E E G amplitude and pulse occurrence sampled at a high rate (e.g. 1 ms) over an extended time span (e.g. 100s) yields an E E G
amplitude histogram and the mean pulse rate. Then the empirical probability of cell firing is determined conditional on the E E G amplitude. This pulse probability distribution is an expression of the nonlinear operations of pulse-wave or wave-pulse conversion. Its numerically computed derivative with respect to wave amplitude is a pulse probability density 1 that is equivalent to the nonlinear gain. This gain curve is the focus of interest in this report.
Conditional Pulse Probability Distribution and Density The nonlinear function has been derived experimentally by analysis of the relation of nerve impulses to the E E G in the olfactory system of cats and rabbits. Details have already been published (Freeman, 1975). Units were recorded singly or in clusters from a microelectrode in cortex p(t) while the E E G was recorded from a surface electrode v(t). After appropriate amplification and filtering of the two signals, the experimental pulse probability conditional on E E G amplitude and time lag was calculated as follows. The E E G was band-pass filtered to remove unit activity and the low-frequency respiratory wave of the olfactory E E G and was then digitized over a time segment 1 Confusion should be avoided between pulse density p and conditional pulse probability density dP(V)/dV The pulse density denotes the pulse rate/unit volume of tissue enclosing a neural subset. It can be estimated by ensembleaveraging over unit trains as in constructing a post-stimulus time histogram. The pulse probability density is estimated by constructing the pulse probability distribution conditional on amplitude V and then computing the derivative with respect to V
239 1. 0
-i.
3
-7
-i WAVE
0 I AMPLITUDE
7
3
~'~-3
-7
-I WAVE
0 i AMPI_ITUDE
2
3 WAVE
AMPLI~UOE
Fig. 2. Three examplesare shown of the normalizedconditional pulse probabilitydistribution (z~)and density([B)fromwakingrabbits; no clear upper asymptotes are seen. The fitted curves (see legend for Fig. 1) have the parameters:
left middle right
Q. 4.84 6.83 14.86
cr 0,33 0.56 0.52.
Ap 0.97 0.91 1.62
Av 0.77 0.74 1.05
of 10-100s; the N values were normalized to zero mean and unit standard deviation, a. The amplitude range was divided into 60 bins 0.1 in width from + 3 to - 3. For each occurrence of an amplitude V at time t the question was asked, did a pulse occur at each time lag T in p(t + T), - 25 ms < T < + 25 ms ? If so, a count of 1 was added to the appropriate bin of a table over and T. The resulting histogram was divided by N and divided by the amplitude probability density to give the conditional pulse probability/5(f/,, T). Cross sections of the table at fixed ~'+ 0 gave/3(T), which was oscillatory at the frequency of the EEG. Cross sections a t T = Tmax where the value of P(T) was maximal gave P(V), which was normalized by dividing the function by the mean firing rate ~. The main conclusions from analysis of experimental curves from the olfactory bulb in anesthetized animals were as follows. a) The maximally varying nonlinear functions were found at Tmax values for which the average of Tmax was equal to one quarter cycle of the dominant frequency of the EEG. This reflected the fact that the E E G was generated by inhibitory neurons and the pulses by excitatory neurons forming a negative feedback loop in the bulb. b) For low amplitude EEG activity/3(f/) at T = Tin,X was nearly linear over the interval + 3o-. c) During high amplitude EEG activity the curves became nonlinear with (on the average) monotonically increasing pulse probability P with increasing wave amplitude and horizontal asymptotes (a "sigmoid" curve as shown in Fig. 1). The sigmoid curve was interpreted to show wave-pulse conversion (wave being the independent variable).
d) The sigmoidal curves were asymmetric about the mean firing rate ~, with a lower asymptote at/3 = 0 and an upper asymptote at 3Po. Analysis of the numerically computed derivatives of the experimental curves (Sunday and Freeman, 1975) showed that e) the dominant peak of the pulse probability density usually occurred in the interval V=(a/2, a), and its magnitude increased linearly with ~. f) The experimental curves from waking rabbits, in which EEG amplitudes were much greater than in anesthetized animals, revealed with increasing EEG amplitude a marked increase in the steepness of the nonlinear curve, the degree of asymptotic asymmetry, and the displacement of the peak of the density curve toward higher amplitude (Fig. 2). The newer findings e)-f) were incompatible with an earlier model (Freeman, 1975), in which it was assumed that the maximal nonlinear gain was at v = 0 , and that the degree of asymptotic asymmetry was fixed at 2, so the construction of a new model was undertaken.
Derivation of the Model
The element of cortical function is taken to be an ensemble of neurons in a local neighborhood that can be lumped without regard to spatial dimensions. The ensemble is described in terms of its topology (the network of connections) and the operations performed by the subset of neurons at each node in the network. The operations of the subset occur in parallel at the same time and are the statistical averages of the operations of the neurons comprising the subset. This report deals with the operations; the t o p o l o g y need not be considered here.
240 M
/3
In the steady state wherein dp/dt = 0 and du/dt = 0, m = m o, u=u o and P=Po. The variables for computation and experimental measurement are
P
v ~- f l ( u -
uo),
q&P-Po,
t.~ c:3
Po
I O
I
i
(1.1)
where & means "is defined as equal to" and fl is a scaling factor. The dimensionless normalized variables are
z Ii20
(1.0)
0
V
u
WAVE AMPLITUDE
Fig. 3. Normalizedpulse densityQ is a functionof wave amplitude u. The steady state values are u=u o and P=Po. The normalized variables.are q =P-Po and v=u-u o. The interveningvariable m is orthogonal to p and u but is shown in the p-u plane for convenience. Curve M is from Eq. (2.2). Curve Q is from Eq. (5.0)
Q & (P- Po)/Po, m & ( m - mo)/m o .
G,(v)F ~(p)Ga(p)F d(v) . When, in the physiological range of operation, the axonal nonlinearity G,(v) dominates and thereby restricts the dendritic operation Ga(p) to a linear range, Gd(p) can be replaced by a constant (. Then the two linear operations can be combined into a single operator F(p) in the sequence
G~(v)(F(p). The linear operation (F(p) need not be considered further here. The proposed model of the nonlinearity contains three state variables. The first two p and u (Fig. 3) represent activity density respectively in the pulse and wave modes. Alternatively q and v are normalized from p and u to zero in the steady state and are used for observation, measurement and computation. The third variable m is to be defined.
(1.3)
Three properties are assigned to the neural subset in the local ensemble. First it is assumed that the m variable is regeneratively voltage-dependent. Corresponding to this property,
dm = c~m. du
(2.0)
For any change in u from u o and m from m o u
=~ j du. Each subset has two state variables : activity density in the pulse density mode p(t) in axons and activity density in the wave mode v(t) in dendrites. There are four operations. Dendritic activity is converted to axonal activity at trigger zones by a nonlinear operation Ga(v). It is linearly delayed and dispersed in time by axonal propagation and multiplied by axonal branching Fa(p). Axonal activity is re-converted to dendritic activity at synapses by a nonlinear operation Ga(p). The synaptically induced activity is integrated and delivered to the trigger zones by a linear operation Fd(v ). In the order of operation the sequence is
(1.2)
(2.1)
UO
From Eqs. (1.0) and (1.3) M = e ~ - 1.
(2.2)
Second, it is assumed that p increases with increasing m, but that there exists an upper limit on pulse density p,, for any given steady state P0 and u o. There are many factors in this limitation, some (such as voltage-dependent saturation of Na-activation, potassium conductance and sodium inactivation) depending on membrane potential, others (such as hyperpolarizing after-potentials) depending on previous activity, and these factors are expressions of nonlinear and time-varying relationships. Over the subset and over the longer time scale of its operation (compared to the time scale of pulses) it is assumed that the nonlinearities become static, and that the rate of increase in pulse density dp/dm is proportional to the difference between p and the prevailing maximum p,, and is inversely proportional to the range P0-P,,.
dp _fl, p~,-p dm P,,- Po'
(3.0)
where fi' is a scaling factor. Integration from Po to p and from m o to m,
fl' " ~voP~--P Pm~PoJo dm' P
dp
_
(3.1)
gives
p=pO+(pm--Po)(1--e--e'(m--mo)/(Pm--Po)).
(3.2)
241 To simplify the notation the prevailing maximal pulse density Pm is expressed as a dimensionless number from Eq. (1.2). (3.3)
Qm ~ (pro- Po)/po .
From Eqs. (1.1), (1.3) and (3.2) Q=Q.,ntl - e
(3.4)
Qmno).
The third property of the system is taken from the empirical observation that for most neurons over a physiological range the steady state pulse rate is proportional to steady state transmembrane current or level of depolarization (Granit et al., 1963 ; Stein, 1967; Calvin, 1975; Heyer and LlinAs, 1976). Po =~Uo,
(4.0)
where 7 is a scaling factor. This linear relationship cannot be derived from the Hodgkin-Huxley system (Agin, 1964 ; Cooley and Dodge, 1966), although it can be extracted by appropriate modifications of the system (Dodge, 1972; Fohlmeister et al., 1974). The relationships between m o and both Po and u o must also be specified. Intuitively it seems that m o should increase monotonically with u o over the physiological range of operation of a neural subset, and P0 should increase monotonically with mo. Equation (4.0) requires that each monotonic function be the inverse of the other. In the absence of experimental or theoretical evidence to the contrary, it is assumed that (4.1)
Po = fi'mo.
The derivative of Eq. (5.0) is equivalent to the experimental pulse probability density curve. It is found by treating the two implicit operations [(2.0) and (3.0)] as serial dQ
d M dQ
dv
dv dM"
(6.0)
From Eqs. (2.2) and (3.4)
dQ
dv = exp [v - (e~- 1)/Qm].
(6.1)
At V=Vo=0 , Qo = 0 ,
dQ dv
(6.2)
- 1.
(6.3)
Equation (6.1) describes the forward gain in the operation Ga(v ). The gain approaches zero as v approaches A _+ oo and goes to a maximum at v=%. The second derivative d~v dvv =
(1-e~/Qm)'
(6.4)
is zero at v = v0, so that (6.5)
v o= ln(Q,, ) .
The peak of the gain curve shifts to the right and its height increases with increased Qm and [-by Eq. (4.4)] u o. The third derivative is
Since m is not measured, fi'= 1. The relationship between u o and Qm is based on the empirical property that p > 0 and therefore Q > - 1. The value for u = 0 is set at Q = - 1 . F r o m Eqs. (1.0), (2.2), and (3.4)
&2 Td =
m = e-P'~ _ 1,
When v =0, there is a maximum in the second derivative for
(4.2)
1
e -MI~ = 1 -~- - Qm"
(4.3)
1
,44
On combining Eqs. (2.2), (3.4), (4.0), and (4.4) (see Fig. 3)
e v = Qm(1.5 + ]/5/2).
Q = - 1 , v = < - u o.
(5.0)
(6.7)
(6.8)
This value for Qm provides a convenient dividing point between high and low sensitivity. For Qm 0 is negative, whereas for Q,, > 2.62 the acceleration is positive. The nonlinear gain for wave-pulse conversion is dp
(2 = Qm(1 - exp[-- (e~ - 1)/Qm]) , v > - uo,
(6.6)
and is zero at
Q,, =2.62.
Because u o is not measured, fl = 1 and
uo ,hi,
(1- 3e~176
d~ = ?u~ exp(v - e - @- 1)/Qm),
(6.9)
and the maximal gain at v = vg is
From Eqs. (1.2) and (4.0) the pulse output is p =TUo( Q + 1).
(5.1)
dv o = T u ~
l + lle"
(6.10)
242 16, 14
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