neuroscience - Neurons with thousands of connections: where are the extra connections coming from?

If every neuron has only one Axon but can can have thousands of (or let's say, even just ten) incoming Axon connections via its dendrites, where are the extra connections coming from?

It seems to me that since neurons dont act as initiators (that is, decide to initiate signals) but rather act more as repeaters (that is, sending incoming signals forward to the neuron that their axon is connect to), then for every neuron, there must be at least one neuron whose axon is connected to it, and at least one neuron that its axon connects to (because without those two things, the neuron does basically nothing, right?), but since each neuron can only have one axon, I'm confused about where all of these extra axons are coming from.

Based on that understanding: 1 billion neurons means exactly 1 billion axons and one billion required inputs (axon -> dendrite connections), eliminating the possibility of any one neuron having more than one dendrite connection, since there can only be one billion axons to connect to those dendrites (and each neuron would require at least one input to function, right?).

But regardless, I've been hearing people talk about neurons having thousands of connections. And based on my understood model, the brain would have to be just one long linear chain of neurons, which makes no sense at all.

So clearly I've got something wrong in my understanding of neural structure. What is it?

Answer

As you say, a neuron can have thousands of inputs via its thousands of dendrites. Each of those dendrites can have a synaptic connection to the (axonic) output of a different neuron. So the neuron can take inputs from thousands of different neurons, not just from one other neuron.

At the other end of the neuron, the output of the axon can form synaptic connections with many distinct neurons; that's what the branching at the end of the axon is for.

Also, it's misleading to say neurons don't "decide to initiate signals". Neurons in the brain do "decide" when to initiate an output. The decision is a complicated function of the arrival of pulses at dendrites from different neurons. Only some combinations of incoming pulses will cause the neuron to fire a pulse on its output axon while other combinations will not. The specific combinations can change over time; this forms the basis of learning.

So a brain neuron is in the middle of a very complicated network of synaptic connections from many neurons to many other neurons.