2006年10月17日 - δ是希腊字母Δ的小写形式，读作delta 在定义函数极限的时候对于任意的ε>0 存在δ>
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偏微分里面的δ怎么念_物理吧_百度贴吧
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Vortex Model of the Brain:
- The Missing Link in Brain Science? ---------------
Vortices in brain activity: their mechanism and significance for perception.
Abstract
Brains interface with the world through perception. The process extracts information from microscopic sensory inputs and incorporates it into the mesoscopic memory store for retrieval in recognition. The process requires creation of spatiotemporal patterns of neural activity. The construction is done through phase transitions in cortical populations that condense the background activity through spontaneous symmetry breaking. Large-scale interactions create fields of synaptically driven activity that is observed by measuring brain waves (electrocorticogram, ECoG) and evaluated by constructing a mesoscopic vectorial order parameter as follows. The negative feedback among excitatory and inhibitory neurons creates spatially and spectrally distributed gamma oscillations (20-80 Hz) in the background activity. Band pass filtering reveals beats in ECoG log analytic power. In some beats that recur at theta rates (3-7 Hz), the order parameter transiently approaches zero, giving a null spike in which the microscopic activity is uniformly disordered (symmetric). A phase transition that is manifested in an analytic phase discontinuity breaks the symmetry. As the null spike terminates, the resurgent order parameter imposes mesoscopic order seen in spatial patterns of ECoG amplitude modulation (AM) that actualize and update the memory of a stimulus. Read-out is through a divergent/convergent projection that performs on cortical output an irreversible spatiotemporal integral transformation. The ECoG reveals a conic phase gradient that accompanies an AM pattern. The phase cone manifests a vortex, which is initiated by the null spike, and which is inferred to help stabilize and prolong its accompanying AM pattern that might otherwise be rapidly degraded by the turbulent neural noise from which it emerges.
assemblies.
g
D
D
D
D
D
Elaboration of the Hypothesis
Entropy-Vortex Wave and Cortical Information Processing
Axiomatic Basis I indicates that the shape of the human brain is not accidental. Rather, it
possesses specific physical meaning in the creation of a virtual sphere in accordance to
convective flow. An important consequence of this fact is that it ensures equivalence of any of
the columns within the entire cortex in an orientation perpendicular to the surface of the brain
and a steady flow in the direction from the core to the surface (Layer VI to I) in all the columns
(Figure 3).
To demonstrate the functional significance of such an organization, here, the effects of a
minor perturbation on such a steady flow will be elaborated (Faber, 1995, Landau & Lifshitz,
1987).
Euler’s equation is given by:
+ ( ⋅∇) + 1 ∇ = 0
∂
∂ p
t
δ
ρ
δv v δv
where δv and δp represent small perturbations in velocity and pressure, respectively.
Similarly, conservation of entropy and the equation of continuity give:
+ ⋅∇ = 0
∂
∂ s
t
δs v δ
+ ⋅∇ + 2∇⋅ = 0
∂
∂δ v δp ρc δv
t
p
where s
c s
p
pδρ δ ( ρ ) δ 2 ∂
= + ∂ and c represent sound velocity.
For a perturbation having the form exp[ik ⋅r − iωt] , one gets:
( ) 0
( ) 0
( ) 0
⋅ − + 2 ⋅ =
⋅ − + =
⋅ − =
v k k v
v k v k
v k
ω δ ρ δ
ρ
ω δ δ
ω δ
p c
p
s
This result prescribes that there will be two types of perturbations, namely, entropy-vortex
wave and sound wave as defined below.
Entropy-Vortex Wave
0
( )
0
0
⋅ =
=
=
≠
= ⋅
k v
v k
δ
δ
δ
δρ δρ
δ
δ
ω
s
s
p
s
p
∇.δv = ik .δv ≠ 0
Sound Wave
0
( )
0
( )
2
2
2 2 2
. =
− ⋅ = ⋅
=
=
− ⋅ =
k v
v k k v
v k
δ
ω δ ρ δ
δ δρ
δ
ω
p c
p c
s
c k
For the purpose of brain modeling, the following points should be emphasized: (1) for
entropy-vortex wave, δs ≠ 0 and ω = v ⋅k ; and (2) for sound wave, δs = 0. These conditions
predict that a perturbation can produce
entropy changes (δs ≠ 0) and, hence,
information processing (Cover & Thomas,
1991, Arbib, 1995) and create
entropy-vortex waves. The fact ω = v ⋅k
ensures that an entropy-vortex wave, which
carries newly processed information,
travels only in the direction identical to the
original flow. In the case of the brain
where the original flow is always
perpendicular to the surface of the brain,
this flow is in the direction parallel to the
columnar arrangement from layer VI to I
(Figure 5). The velocity of the sound wave
is subsonic and, hence, will travel in all
directions3. However, a sound wave does
not carry any new information (δs = 0).
3 Sound waves
assemblies.
g
D
D
D
D
D
Elaboration of the Hypothesis
Entropy-Vortex Wave and Cortical Information Processing
Axiomatic Basis I indicates that the shape of the human brain is not accidental. Rather, it
possesses specific physical meaning in the creation of a virtual sphere in accordance to
convective flow. An important consequence of this fact is that it ensures equivalence of any of
the columns within the entire cortex in an orientation perpendicular to the surface of the brain
and a steady flow in the direction from the core to the surface (Layer VI to I) in all the columns
(Figure 3).
To demonstrate the functional significance of such an organization, here, the effects of a
minor perturbation on such a steady flow will be elaborated (Faber, 1995, Landau & Lifshitz,
1987).
Euler’s equation is given by:
+ ( ⋅∇) + 1 ∇ = 0
∂
∂ p
t
δ
ρ
δv v δv
where δv and δp represent small perturbations in velocity and pressure, respectively.
Similarly, conservation of entropy and the equation of continuity give:
+ ⋅∇ = 0
∂
∂ s
t
δs v δ
+ ⋅∇ + 2∇⋅ = 0
∂
∂δ v δp ρc δv
t
p
where s
c s
p
pδρ δ ( ρ ) δ 2 ∂
= + ∂ and c represent sound velocity.
For a perturbation having the form exp[ik ⋅r − iωt] , one gets:
( ) 0
( ) 0
( ) 0
⋅ − + 2 ⋅ =
⋅ − + =
⋅ − =
v k k v
v k v k
v k
ω δ ρ δ
ρ
ω δ δ
ω δ
p c
p
s
This result prescribes that there will be two types of perturbations, namely, entropy-vortex
wave and sound wave as defined below.
Entropy-Vortex Wave
0
( )
0
0
⋅ =
=
=
≠
= ⋅
k v
v k
δ
δ
δ
δρ δρ
δ
δ
ω
s
s
p
s
p
∇.δv = ik .δv ≠ 0
Sound Wave
0
( )
0
( )
2
2
2 2 2
. =
− ⋅ = ⋅
=
=
− ⋅ =
k v
v k k v
v k
δ
ω δ ρ δ
δ δρ
δ
ω
p c
p c
s
c k
For the purpose of brain modeling, the following points should be emphasized: (1) for
entropy-vortex wave, δs ≠ 0 and ω = v ⋅k ; and (2) for sound wave, δs = 0. These conditions
predict that a perturbation can produce
entropy changes (δs ≠ 0) and, hence,
information processing (Cover & Thomas,
1991, Arbib, 1995) and create
entropy-vortex waves. The fact ω = v ⋅k
ensures that an entropy-vortex wave, which
carries newly processed information,
travels only in the direction identical to the
original flow. In the case of the brain
where the original flow is always
perpendicular to the surface of the brain,
this flow is in the direction parallel to the
columnar arrangement from layer VI to I
(Figure 5). The velocity of the sound wave
is subsonic and, hence, will travel in all
directions3. However, a sound wave does
not carry any new information (δs = 0).
3 Sound waves
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