Analytical considerations

Under which conditions do oscillations induce a transition from a regime of non-robust to ro-bust synchrony propagation? In particular, what is the (minimal) amplitude or the degree of synchrony required that allow for robust signal propagation?

To answer this question, we investigate the emergence of oscillation-supported propagation in FFNs with non-additive couplings analytically. We employ a self-consistency approach (cf. also methods introduced in Memmesheimer and Timme, 2012; Jahnke et al., 2012, 2013) to derive an

approximation of the iterated map for the average size of a synchronous pulse that propagates along the layers of an FFN. In particular, we find a scaling law for the amplitude of the external oscillations that enable stable propagation as a function of the system parameters and the dendritic nonlinearity.

Synchronous spiking of neurons in some layer causes a synchronous input to the neurons of the next layer. In the presence of oscillations of suitable frequency, e.g., ν^s ≈ν^nat, this input may be supported by inputs from the external oscillations. Then the total excitatory input

I =Ie+Ic (6.17)

is the sum of inputs arising from external oscillations, Ie, and from the preceding layer, Ic. In networks with non-additive coupling, the spiking probability p^sp due to a synchronous input I below the dendritic threshold Θb is typically much smaller than due to a suprathreshold input (cf. Figure 6.3a). We thus assume that only neurons that receive a suprathreshold input (I >Θb) generate a spike with fixed probability p^∗, i.e.,

p^sp(I) :=

(p^∗ ifI ≥Θb

0 ifI <Θb

. (6.18)

Thus, neurons process synchronous signals like simple threshold units, i.e., they generate no response for subthreshold inputs and a fixed response for suprathreshold inputs. For clarity of presentation, we assume that the firing probability p^∗ is fixed. In general it might be reduced by inhibitory input, but the extension is straightforward and leads to similar results (cf. also Jahnke et al., 2014a).

The timing of somatic spikes initiated by dendritic spikes is highly precise, i.e., the temporal distribution of somatic spikes triggered by dendritic ones is very narrow (cf. Figure 6.4a), in the sub-millisecond range (cf. also Ariav et al., 2003). In particular, the jitter in time is typically much smaller than the dendritic integration window ∆T^s. This let us assume that a synchronous pulse packet in one layer causes synchronous spiking within a time interval smaller than ∆T^s in the next layer and so on.

In the following we calculate the probability density functionf_I(I) for the total excitatory input I to the neurons of a given layer conditioned on (i) the number of synchronously spiking neurons in the previous layer, gⁱⁿ, and (ii) the amplitude of external oscillations, N_e. Then, the average number of synchronously spiking neurons in the considered layer is

g^out =ω Z ∞

0

p^sp(I)fI(I|gⁱⁿ, Ne)dI (6.19)

=ωp^∗ Z ∞

Θ_b

f_I(I|gⁱⁿ, N_e)dI. (6.20)

First we consider the input from the previous layer. Given the random topology of the FFN, the probability that a neuron receives exactlyk(out of the maximal numbergⁱⁿ) inputs is binomially distributed,

p(k) = gⁱⁿ k

!

(pex)^k(1−pex)^gin−k. (6.21)

For a sufficiently large number gⁱⁿ of neurons participating in the synchronous pulses, we can approximate the binomial distribution (6.21) by a Gaussian distribution and thus the excitatory synchronous input follows

I_c=kε_c∼ Nµ_c, σ_c² (6.22)

with mean

µ_c=ε_cgⁱⁿp_ex (6.23)

and standard deviation

σ_c=ε_c^qgⁱⁿp_ex(1−p_ex). (6.24) Likewise, the number of excitatory inputs l each neuron receives within one oscillation period from the external (virtual) neuron population is binomially distributed, l ∼ B Ne, p^ext_ex (cf.

Section 6.2.6). The arrival times are drawn from a Gaussian distribution with standard deviation σ^s. We assume that propagation of synchrony in the FFN and the external oscillations are in-phase. Then forσ^s >0 the fraction

p_∆T^s =^Z

∆Ts 2

−^∆T₂^s

√1

2πσ^sexp

"

−1 2

τ σ^s

2#

dτ (6.25)

= Erf∆T^s

√8σ^s

(6.26) of the additional inputs arrive within the dendritic integration window ∆T^s and can support the generation of dendritic spikes. For σ^s = 0, all inputs are received synchronously and thus p_∆T^s = 1; for non-zero σ^s the effective size of the external oscillation (i.e., the effective average number of neurons that may contribute to the generation of dendritic spikes) is

N_e^eff=p_∆T^sNe, (6.27)

and the number of excitatory inputs from the oscillatory neuron population is distributed ac-cording tol∼BN_e^eff, p^ext_ex .

For sufficiently largeN_e^eff, one can again use a Gaussian approximation which yields

Ie∼ Nµe, σ_e² (6.28)

with

µ_e=ε^ext_p N_e^effp^ext_ex and σ_e=ε^ext_p q

N_e^effp^ext_ex (1−p^ext_ex ). (6.29) The sum of the inputsI_e and I_c is then also approximately Gaussian distributed,

I =Ie+Ic∼ Nµ, σ², (6.30)

0 50 100 150

0 50 100 150

0 50 100 150 0 50 100 150

g^out g^out

in

gin g

ε =_c 1.0 nS ε =_c 1.5 nS ε =_c 2.5 nS

N = 0_e N = 204_e N = 383_e

ε =c 1.0 nS N = 0_e

(a) (b)

Figure 6.10: Map yielding the temporal evolution of the average size of a synchronous pulse in an FFN with non-additive coupling(cf. Equation 6.33;ω= 200,p_ex= 0.05, Θb= 8.65nS). (a) N_e =0 (absence of external oscillations), different colors indicate different coupling strengthε_c. (b) The coupling strengthε_c= 1.0nS is fixed and external oscillations (p^ext_ex = 0.05, ε^ext_p = 0.3nS, σ^s = 0ms) are present, different colors indicate different N_e. With increasing (a) connection strengthε_c or increasing (b) oscillation amplitudeN_e, two fixed points emerge by a tangent bifurcation. This bifurcation point marks the transition from a regime where no propagation is possible to a regime where persistent propagation of synchrony can be achieved (cf. also Figure 6.3b).

with mean µ=µe+µc and varianceσ²=σ_e²+σ_c², i.e.,

µ=ε^ext_p N_e^effp^ext_ex +ε_cgⁱⁿp_ex (6.31) and

σ = r

ε^ext_p ²N_e^effp^ext_ex (1−p^ext_ex ) +ε²_cgⁱⁿp_ex(1−p_ex). (6.32) Using the distribution (6.30) of I allows us to specify the iterated map for the average size of a synchronous pulse according to Equation (6.20),

g^out= ωp^∗ 2

1 + Erfµ√−Θb

2σ

, (6.33)

where the size of the initial pulse packetgⁱⁿappears as argument ofµandσ (see Equations 6.31 and 6.32).

As explained in Section 6.3.1 (cf. Equation 6.10), the fixed points G^∗ =g^out =gⁱⁿ of Equation (6.33) determine the stability of the propagation of a synchronous pulse. With increasing cou-pling strength two fixed points emerge via a tangent bifurcation (Figure 6.10a; cf. also Figure 6.3b), and external oscillations have a similar effect (Figure 6.10b). This transition enables robust propagation of synchrony, and the external oscillations thus reduce the critical connec-tion strength ε^∗_NL (i.e., the minimal coupling strength for which robust signal propagation is possible).

For a given network setup, N_e=N_e^∗ specifies the minimal size of the external oscillation which enables stable propagation of synchrony. It can be found by numerically determining the bifur-cation point of Equation (6.33). Additionally, one can derive a scaling law for N_e^∗ based on two observations:

1. In the absence of external oscillations (N_e = 0), the position of the bifurcation point of Equation (6.33) depends on the coupling strengthεcand the dendritic threshold Θb only via the quotient

κ:= Θb

ε_c, (6.34)

which is the number of spikes from the preceding layer that are needed to elicit a dendritic spike. Equation (6.33) reads

g^out= ωp^∗

2 1 + Erf

"

gⁱⁿp_ex−κ p2gⁱⁿp_ex(1−p_ex)

#!

. (6.35)

For a given network setup, the connection probabilityp_ex, group sizeω and spiking prob-ability p^∗ (which is determined by the ground state and the parameters of the dendritic spike) are fixed. Thus the bifurcation point where the fixed points G^∗₁ =G^∗₂ =g^out =gⁱⁿ appear by a tangent bifurcation, depends solely onκ (the only unknown quantity). Con-sequently, there is someκ^∗ =κ specifying this bifurcation point, i.e., the transition point from non-propagating to propagating regime depends just on the number of spikes nec-essary to elicit a dendritic spike. The actual value κ^∗ can be found either by numerical simulation of the system, numerical solution of Equation (6.35) or by the analytical meth-ods introduced in Jahnke et al. (2012, 2013).

2. The main influence of external oscillatory inputs is an effective reduction of the dendritic threshold Θb, such that the properties of the system described above can be approximated by a network without external oscillatory input, but with a reduced dendritic threshold Θ^eff_b < Θb: In the setups considered the additional oscillatory input contributes to the generation of dendritic spikes, but the main contribution arises from the input arriving from the previous layer (the signal to be propagated), i.e., µe < µc. Moreover, the feed-forward connections ε_c are enhanced compared to the remaining excitatory couplings, ε^ext_p < εc. Thus the total variation of the input σ = σ_e² +σ²_c (cf. Equation 6.32) is dominated by the contributionσ_c² of the input from the previous layer,

ε^ext_p µ_e1−p^ext_ex < ε_pµ_c(1−p_ex) (6.36)

σ_e²< σ²_c. (6.37)

In particular for ε^ext_p ε_c the contribution of the external inputs to the total variation of the input becomes negligible, i.e.,σ²_e σ²_c, and the argument of the error function in Equation (6.33) simplifies to

µ√−Θb

2σ = µc+µe−Θb

p2 (σ²_c+σ²_e) (6.38)

≈ µ_c√−Θ^eff_b

2σ_c (6.39)

where we defined the effective dendritic threshold

Θ^effb := Θb−µe. (6.40)

The above observations indicate that the bifurcation point is found for some constant κ^∗ = Θ^eff_b

εc = Θb−µ_e εc

, (6.41)

such that the minimal size of the external oscillations N_e^∗, which enables propagation of syn-chrony, is given by (using Equations 6.29, 6.27 and 6.26)

N_e^∗= Erf∆T^s

√8σ^s

−1Θb−ε_cκ^∗

ε^ext_p p^ext_ex . (6.42)

Equation (6.42) indicates that N_e^∗ changes linearly with the coupling strength εc (cf. Figure 6.5 and 6.6). Further it is inversely proportional to the coupling strength between the external oscillatory population and the neurons of the FFN, N_e^∗ ∝1/ε^ext_p , and the dependence of N_e^∗ on the temporal spread σ^s of the external oscillations is determined by the prefactor 1/p_∆T^s. The above results are derived for isolated FFNs. However, we will show and discuss in the following Appendix 6.5.3 that the results hold in good approximation also for FFNs that are part of recurrent networks.

Im Dokument Neural Networks with Nonlinear Couplings (Seite 149-154)