We consider the effect of distributed delays in neural feedback systems. The avian optic tectum is reciprocally connected with the nucleus isthmi. Extracellular stimulation combined with intracellular recordings reveal a range of signal delays from 4 to 9 ms between isthmotectal elements. This observation together with prior mathematical analysis concerning the influence of a delay distribution on system dynamics raises the question whether a broad delay distribution can impact the dynamics of neural feedback loops. For a system of reciprocally connected model neurons, we found that distributed delays enhance system stability in the following sense. With increased distribution of delays, the system converges faster to a fixed point and converges slower toward a limit cycle. Further, the introduction of distributed delays leads to an increased range of the average delay value for which the system's equilibrium point is stable. The enhancement of stability with increasing delay distribution is caused by the introduction of smaller delays rather than the distribution per se.
Deep Dive into Distributed delays stabilize neural feedback systems.
We consider the effect of distributed delays in neural feedback systems. The avian optic tectum is reciprocally connected with the nucleus isthmi. Extracellular stimulation combined with intracellular recordings reveal a range of signal delays from 4 to 9 ms between isthmotectal elements. This observation together with prior mathematical analysis concerning the influence of a delay distribution on system dynamics raises the question whether a broad delay distribution can impact the dynamics of neural feedback loops. For a system of reciprocally connected model neurons, we found that distributed delays enhance system stability in the following sense. With increased distribution of delays, the system converges faster to a fixed point and converges slower toward a limit cycle. Further, the introduction of distributed delays leads to an increased range of the average delay value for which the system’s equilibrium point is stable. The enhancement of stability with increasing delay distribut
arXiv:0712.0036v1 [physics.bio-ph] 3 Dec 2007
Distributed delays stabilize neural feedback systems
Ulrike Meyer,1 Jing Shao,2 Saurish Chakrabarty,2 Sebastian F. Brandt,2 Harald Luksch,3 and Ralf Wessel2, ∗
1Institute for Biology II, RWTH, 52074 Aachen, Germany
2 Department of Physics, Washington University in St. Louis, MO 63130-4899, USA
3Institute for Zoology, Technical University Munich, 85350 Freising-Weihenstephan, Germany
(Dated: November 30, 2007)
We consider the effect of distributed delays in neural feedback systems. The avian optic tectum
is reciprocally connected with the nucleus isthmi. Extracellular stimulation combined with intra-
cellular recordings reveal a range of signal delays from 4 to 9 ms between isthmotectal elements.
This observation together with prior mathematical analysis concerning the influence of a delay dis-
tribution on system dynamics raises the question whether a broad delay distribution can impact the
dynamics of neural feedback loops. For a system of reciprocally connected model neurons, we found
that distributed delays enhance system stability in the following sense. With increased distribution
of delays, the system converges faster to a fixed point and converges slower toward a limit cycle.
Further, the introduction of distributed delays leads to an increased range of the average delay value
for which the system’s equilibrium point is stable. The enhancement of stability with increasing
delay distribution is caused by the introduction of smaller delays rather than the distribution per se.
PACS numbers: 87.19.L-, 87.18.Sn, 87.10.-e
I.
INTRODUCTION
The signal flow in the brain is not just feedforward;
rather, feedback dominates most neural pathways [1]. Of-
ten pairs of reciprocally connected neurons are spatially
separate by several millimeters. For instance, the primate
corticothalamic feedback loop extends over a distance of
approximately 100 mm. Thus, for a typical action po-
tential speed of 1 mm/ms we expect a signal delay of
100 ms. When signal delays are larger than the neural
response time, complex loop dynamics emerge [2, 3, 4].
For reciprocally connected populations of neurons,
large delays can introduce another dimension, namely the
distribution of delay times. Such a distribution could be
an epiphenomenon in the evolution of larger brains, or
it could be of adaptive significance. Work from applied
mathematics states an influence of the distribution of de-
lay times on system dynamics [5, 6, 7, 8, 9, 10]. Intrigued
by the latter possibility, we asked two questions: What is
the distribution of delay times in an experimentally ac-
cessible neural feedback system? What is the impact of
distributed delays on a mathematically tractable neural
model feedback system?
We measured the distribution of delay times in the
isthmotectal feedback system of birds [Fig. 1(a)] [11, 12].
The avian isthmic nuclei (parabigeminal nucleus in mam-
mals) receive a topographically organized projection from
the tectum (superior colliculus in mammals), to which
they project back and have been conjectured to medi-
ate spatiotemporal attentional mechanisms [13, 14, 15].
The isthmic nuclei in birds consist of three substructures:
pars parvocellularis (Ipc), pars magnocellularis (Imc),
and pars semilunaris (SLu) that are spatially separated
∗Electronic address: rw@physics.wustl.edu
from the tectum [16, 17]. In response to visual stimu-
lation, the Ipc neurons undergo a transition from quies-
cence to rhythmic firing [15, 18]. Delays can drive a neu-
ral feedback loop over a stability boundary resulting in
oscillatory behavior [19, 20, 21, 22, 23, 24]. To elucidate
the impact of a delay distribution on the system dynam-
ics, we investigated, through numerical simulations and
mathematical analysis, a model of reciprocally coupled
neurons with distributed delays.
II.
MEASURED DISTRIBUTION OF DELAYS
To measure the signal delays between pairs of isth-
motectal elements, we obtained intracellular whole-cell
recordings from identified neurons in a midbrain slice
preparation and stimulated groups of presynaptic neu-
rons or axons with brief electrical pulses delivered extra-
cellularly [Fig. 1(b)].
Neurons were identified by their
location within the midbrain slice preparation and for a
subset of recorded neurons we obtained additional iden-
tification via intracellular fills [16, 17].
A subpopulation of tectal layer 10 (L10) neurons
project to both the ipsilateral Ipc and Imc in a topo-
graphic fashion [16, 17, 25, 26, 27, 28].
Their apical
dendrite courses straight up to layer 2 with few ramifi-
cations, and basal dendrites reach down to the border of
layer 13. Retinal axon terminals overlap with the api-
cal dendrite in tectal layers 2 to 7 [29, 30]. We placed
a stimulus electrode in layer 2 to 4 (L2-4) and recorded
from L10 neurons with whole-cell recordings in response
to L2-4 stimulation. The delays from the beginning of the
stimulus pulse to the onset of the L10 response ranged
from 4 to 15 ms with a m
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